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Search: a024787 -id:a024787
Displaying 1-10 of 13 results found. page 1 2
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A024786 Number of 2's in all partitions of n. +10
42
0, 1, 1, 3, 4, 8, 11, 19, 26, 41, 56, 83, 112, 160, 213, 295, 389, 526, 686, 911, 1176, 1538, 1968, 2540, 3223, 4115, 5181, 6551, 8191, 10269, 12756, 15873, 19598, 24222, 29741, 36532, 44624, 54509, 66261, 80524, 97446, 117862, 142029, 171036, 205290, 246211 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
Also number of partitions of n-1 with a distinguished part different from all the others. [Comment corrected by Emeric Deutsch, Aug 13 2008]
In general the number of times that j appears in the partitions of n equals Sum_{k<n, k = n (mod j)} P(k). In particular this gives a formula for a(n), A024787, ..., A024794, for j = 2,...,10; it generalizes the formula given for A000070 for j=1. - Jose Luis Arregui (arregui(AT)posta.unizar.es), Apr 05 2002
Equals row sums of triangle A173238. - Gary W. Adamson, Feb 13 2010
The sums of two successive terms give A000070. - Omar E. Pol, Jul 12 2012
a(n) is also the difference between the sum of second largest and the sum of third largest elements in all partitions of n. More generally, the number of occurrences of k in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+1)st largest elements in all partitions of n. And more generally, the sum of the number of occurrences of k, k+1, k+2..k+m in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+m+1)st largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012
Number of singletons in all partitions of n-1. A singleton in a partition is a part that occurs exactly once. Example: a(5) = 4 because in the partitions of 4, namely [1,1,1,1], [1,1,2'], [2,2], [1',3'], [4'] we have 4 singletons (marked by '). - Emeric Deutsch, Sep 12 2016
a(n) is also the number of non-isomorphic vertex-transitive cover graphs of lattice quotients of essential lattice congruences of the weak order on the symmetric group S_{n-1}. See Table 1 in the Hoang/Mütze reference in the Links section. - Torsten Muetze, Nov 28 2019
Assuming a partition is in weakly decreasing order, a(n) is also the number of times -1 occurs in the differences of the partitions of n+1. - George Beck, Mar 28 2023
REFERENCES
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 184.
LINKS
Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.
Manosij Ghosh Dastidar and Sourav Sen Gupta, Generalization of a few results in Integer Partitions, arXiv preprint arXiv:1111.0094 [cs.DM], 2011.
Emeric Deutsch et al., Problem 11237, Amer. Math. Monthly, 115 (No. 7, 2008), 666-667. [From Emeric Deutsch, Aug 13 2008]
Hung Phuc Hoang and Torsten Mütze, Combinatorial generation via permutation languages. II. Lattice congruences, arXiv:1911.12078 [math.CO], 2019.
Joseph Vandehey, Digital problems in the theory of partitions, Integers (2024) Vol. 24A, Art. No. A18. See p. 3.
FORMULA
a(n) = Sum_{k=1..floor(n/2)} A000041(n-2k). - Christian G. Bower, Jun 22 2000
a(n) = Sum_{k<n, k = n (mod 2)} P(k), P(k) = number of partitions of k as in A000041, P(0) = 1. - Jose Luis Arregui (arregui(AT)posta.unizar.es), Apr 05 2002
G.f.: x^2/((1-x)*(1-x^2)^2))*Product_{j>=3} 1/(1-x^j) from Riordan reference second term, last eq.
a(n) = A006128(n-1) - A194452(n-1). - Omar E. Pol, Nov 20 2011
a(n) = A181187(n,2) - A181187(n,3). - Omar E. Pol, Oct 25 2012
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/2) * Pi * sqrt(n)) * (1 - 25*Pi/(24*sqrt(6*n)) + (25/48 + 433*Pi^2/6912)/n). - Vaclav Kotesovec, Mar 07 2016, extended Nov 05 2016
a(n) = Sum_{k} k * A116595(n-1,k). - Emeric Deutsch, Sep 12 2016
G.f.: x^2/((1 - x)*(1 - x^2)) * Sum_{n >= 0} x^(2*n)/( Product_{k = 1..n} 1 - x^k ); that is, convolution of A004526 (partitions into 2 parts, or, modulo offset differences, partitions into parts <= 2) and A002865 (partitions into parts >= 2). - Peter Bala, Jan 17 2021
EXAMPLE
From Omar E. Pol, Oct 25 2012: (Start)
For n = 7 we have:
--------------------------------------
. Number
Partitions of 7 of 2's
--------------------------------------
7 .............................. 0
4 + 3 .......................... 0
5 + 2 .......................... 1
3 + 2 + 2 ...................... 2
6 + 1 .......................... 0
3 + 3 + 1 ...................... 0
4 + 2 + 1 ...................... 1
2 + 2 + 2 + 1 .................. 3
5 + 1 + 1 ...................... 0
3 + 2 + 1 + 1 .................. 1
4 + 1 + 1 + 1 .................. 0
2 + 2 + 1 + 1 + 1 .............. 2
3 + 1 + 1 + 1 + 1 .............. 0
2 + 1 + 1 + 1 + 1 + 1 .......... 1
1 + 1 + 1 + 1 + 1 + 1 + 1 ...... 0
------------------------------------
. 24 - 13 = 11
.
The difference between the sum of the second column and the sum of the third column of the set of partitions of 7 is 24 - 13 = 11 and equals the number of 2's in all partitions of 7, so a(7) = 11.
(End)
MAPLE
b:= proc(n, i) option remember; local f, g;
if n=0 or i=1 then [1, 0]
else f:= b(n, i-1); g:= `if`(i>n, [0$2], b(n-i, i));
[f[1]+g[1], f[2]+g[2]+`if`(i=2, g[1], 0)]
fi
end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50); # Alois P. Heinz, May 18 2012
MATHEMATICA
Table[ Count[ Flatten[ IntegerPartitions[n]], 2], {n, 1, 50} ]
(* Second program: *)
b[n_, i_] := b[n, i] = Module[{f, g}, If[n==0 || i==1, {1, 0}, f = b[n, i - 1]; g = If[i>n, {0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + If[i == 2, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Sep 22 2015, after Alois P. Heinz *)
Join[{0}, (1/((1 - x^2) QPochhammer[x]) + O[x]^50)[[3]]] (* Vladimir Reshetnikov, Nov 22 2016 *)
Table[Sum[(1 + (-1)^k)/2 * PartitionsP[n-k], {k, 2, n}], {n, 1, 50}] (* Vaclav Kotesovec, Aug 27 2017 *)
PROG
(Python)
from sympy import npartitions
def A024786(n): return sum(npartitions(n-(k<<1)) for k in range(1, (n>>1)+1)) # Chai Wah Wu, Oct 25 2023
CROSSREFS
Cf. A066633, A024787, A024788, A024789, A024790, A024791, A024792, A024793, A024794, A173238.
Column 2 of A060244.
First differences of A000097.
KEYWORD
nonn
AUTHOR
Clark Kimberling
STATUS
approved
A024788 Number of 4's in all partitions of n. +10
15
0, 0, 0, 1, 1, 2, 3, 6, 8, 13, 18, 28, 38, 55, 74, 105, 139, 190, 250, 336, 436, 575, 740, 963, 1228, 1577, 1995, 2538, 3186, 4013, 5005, 6256, 7751, 9617, 11847, 14605, 17894, 21927, 26730, 32582, 39531, 47942, 57915, 69920, 84114, 101116, 121176, 145095, 173248 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
COMMENTS
The sums of four successive terms give A000070. - Omar E. Pol, Jul 12 2012
a(n) is also the difference between the sum of 4th largest and the sum of 5th largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012
a(n+4) is the number of n-vertex graphs that do not contain a triangle, P_4, or K_2,3 as induced subgraph. These are the K_2,3-free bipartite cographs. Bipartite cographs are graph that are disjoint unions of complete bipartite graphs [Babel et al. Corollary 2.2], and forbidding K_2,3 leaves one possible component for each size except size 4, where there are two. Thus, this number is A000041(n) + a(n) = a(n+4). - Falk Hüffner, Jan 11 2016
a(n) (n>=3) is the number of even singletons in all partitions of n-2 (by a singleton we mean a part that occurs exactly once). Example: a(7) = 3 because in the partitions [5], [4*,1], [3,2*], [3,1,1], [2,2,1], [2*,1,1,1], [1,1,1,1,1] we have 3 even singletons (marked by *). The statement of this comment can be obtained by setting k=2 in Theorem 2 of the Andrews et al. reference. - Emeric Deutsch, Sep 13 2016
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
G. E. Andrews and E. Deutsch, A note on a method of Erdos and the Stanley-Elder theorems, Integers, 16 (2016), A24.
L. Babel, A. Brandstädt, and V. B. Le, Recognizing the P4-structure of bipartite graphs, Discrete Appl. Math. 93 (1999), 157-168.
David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.
FORMULA
a(n) = A181187(n,4) - A181187(n,5). - Omar E. Pol, Oct 25 2012
From Peter Bala, Dec 26 2013: (Start)
a(n+4) - a(n) = A000041(n). a(n) + a(n+2) = A024786(n).
O.g.f.: x^4/(1 - x^4) * product {k >= 1} 1/(1 - x^k) = x^4 + x^5 + 2*x^6 + 3*x^7 + ....
Asymptotic result: log(a(n)) ~ 2*sqrt(Pi^2/6)*sqrt(n) as n -> inf. (End)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*Pi*sqrt(2*n)) * (1 - 49*Pi/(24*sqrt(6*n)) + (49/48 + 1633*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 05 2016
G.f.: x^4/((1 - x)*(1 - x^2)*(1 - x^3)*(1 - x^4)) * Sum_{n >= 0} x^(4*n)/( Product_{k = 1..n} 1 - x^k ); that is, convolution of A026810 (partitions into 4 parts, or, modulo offset differences, partitions into parts <= 4) and A008484 (partitions into parts >= 4). - Peter Bala, Jan 17 2021
EXAMPLE
From Omar E. Pol, Oct 25 2012: (Start)
For n = 7 we have:
--------------------------------------
. Number
Partitions of 7 of 4's
--------------------------------------
7 .............................. 0
4 + 3 .......................... 1
5 + 2 .......................... 0
3 + 2 + 2 ...................... 0
6 + 1 .......................... 0
3 + 3 + 1 ...................... 0
4 + 2 + 1 ...................... 1
2 + 2 + 2 + 1 .................. 0
5 + 1 + 1 ...................... 0
3 + 2 + 1 + 1 .................. 0
4 + 1 + 1 + 1 .................. 1
2 + 2 + 1 + 1 + 1 .............. 0
3 + 1 + 1 + 1 + 1 .............. 0
2 + 1 + 1 + 1 + 1 + 1 .......... 0
1 + 1 + 1 + 1 + 1 + 1 + 1 ...... 0
------------------------------------
. 7 - 4 = 3
The difference between the sum of the fourth column and the sum of the fifth column of the set of partitions of 7 is 7 - 4 = 3 and equals the number of 4's in all partitions of 7, so a(7) = 3.
(End)
MAPLE
b:= proc(n, i) option remember; local f, g;
if n=0 or i=1 then [1, 0]
else f:= b(n, i-1); g:= `if`(i>n, [0$2], b(n-i, i));
[f[1]+g[1], f[2]+g[2]+`if`(i=4, g[1], 0)]
fi
end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..100); # Alois P. Heinz, Oct 27 2012
MATHEMATICA
Table[ Count[ Flatten[ IntegerPartitions[n]], 4], {n, 1, 50} ]
(* second program: *)
b[n_, i_] := b[n, i] = Module[{g}, If[n == 0 || i == 1, {1, 0}, g = If[i > n, {0, 0}, b[n - i, i]]; b[n, i - 1] + g + {0, If[i == 4, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 09 2015, after Alois P. Heinz *)
CROSSREFS
Cf. A000041, A066633, A024786, A024787, A024789, A024790, A024791, A024792, A024793, A024794.
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling
STATUS
approved
A024789 Number of 5's in all partitions of n. +10
13
0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 12, 17, 25, 35, 50, 68, 94, 126, 170, 226, 299, 391, 511, 660, 853, 1091, 1393, 1766, 2235, 2811, 3527, 4403, 5484, 6800, 8415, 10369, 12752, 15627, 19110, 23298, 28346, 34389, 41642, 50295, 60636, 72929, 87563, 104903, 125470 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,7
COMMENTS
The sums of five successive terms give A000070. - Omar E. Pol, Jul 12 2012
a(n) is also the difference between the sum of 5th largest and the sum of 6th largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.
FORMULA
a(n) = A181187(n,5) - A181187(n,6). - Omar E. Pol, Oct 25 2012
a(n) ~ exp(Pi*sqrt(2*n/3)) / (10*Pi*sqrt(2*n)) * (1 - 61*Pi/(24*sqrt(6*n)) + (61/48 + 2521*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 05 2016
G.f.: x^5/(1 - x^5) * Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Apr 06 2017
EXAMPLE
From Omar E. Pol, Oct 25 2012: (Start)
For n = 8 we have:
--------------------------------------
. Number
Partitions of 8 of 5's
--------------------------------------
8 .............................. 0
4 + 4 .......................... 0
5 + 3 .......................... 1
6 + 2 .......................... 0
3 + 3 + 2 ...................... 0
4 + 2 + 2 ...................... 0
2 + 2 + 2 + 2 .................. 0
7 + 1 .......................... 0
4 + 3 + 1 ...................... 0
5 + 2 + 1 ...................... 1
3 + 2 + 2 + 1 .................. 0
6 + 1 + 1 ...................... 0
3 + 3 + 1 + 1 .................. 0
4 + 2 + 1 + 1 .................. 0
2 + 2 + 2 + 1 + 1 .............. 0
5 + 1 + 1 + 1 .................. 1
3 + 2 + 1 + 1 + 1 .............. 0
4 + 1 + 1 + 1 + 1 .............. 0
2 + 2 + 1 + 1 + 1 + 1 .......... 0
3 + 1 + 1 + 1 + 1 + 1 .......... 0
2 + 1 + 1 + 1 + 1 + 1 + 1 ...... 0
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 .. 0
------------------------------------
. 7 - 4 = 3
The difference between the sum of the fifth column and the sum of the sixth column of the set of partitions of 8 is 7 - 4 = 3 and equals the number of 5's in all partitions of 8, so a(8) = 3.
(End)
MAPLE
b:= proc(n, i) option remember; local g;
if n=0 or i=1 then [1, 0]
else g:= `if`(i>n, [0$2], b(n-i, i));
b(n, i-1) +g +[0, `if`(i=5, g[1], 0)]
fi
end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..100); # Alois P. Heinz, Oct 27 2012
MATHEMATICA
Table[ Count[ Flatten[ IntegerPartitions[n]], 5], {n, 1, 50} ]
(* second program: *)
b[n_, i_] := b[n, i] = Module[{g}, If[n == 0 || i == 1, {1, 0}, g = If[i > n, {0, 0}, b[n - i, i]]; b[n, i - 1] + g + {0, If[i == 5, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 09 2015, after Alois P. Heinz *)
PROG
(PARI) x='x+O('x^50); concat([0, 0, 0, 0], Vec(x^5/(1 - x^5) * prod(k=1, 50, 1/(1 - x^k)))) \\ Indranil Ghosh, Apr 06 2017
CROSSREFS
Cf. A066633, A024786, A024787, A024788, A024790, A024791, A024792, A024793, A024794.
KEYWORD
nonn
AUTHOR
Clark Kimberling
STATUS
approved
A024790 Number of 6's in all partitions of n. +10
13
0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 12, 16, 24, 33, 47, 63, 89, 117, 159, 209, 278, 360, 474, 607, 786, 1001, 1280, 1615, 2049, 2565, 3222, 4011, 4998, 6180, 7653, 9407, 11571, 14154, 17308, 21063, 25630, 31044, 37586, 45339, 54646, 65646, 78804, 94305, 112761, 134473 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,8
COMMENTS
The sums of six successive terms give A000070. - Omar E. Pol, Jul 12 2012
a(n) is also the difference between the sum of 6th largest and the sum of 7th largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.
FORMULA
a(n) = A181187(n,6) - A181187(n,7). - Omar E. Pol, Oct 25 2012
From Peter Bala, Dec 26 2013: (Start)
a(n+6) - a(n) = A000041(n). a(n) + a(n+3) = A024787(n).
a(n) + a(n+2) + a(n+4) = A024786(n).
O.g.f.: x^6/(1 - x^6) * product {k >= 1} 1/(1 - x^k) = x^6 + x^7 + 2*x^8 + 3*x^9 + ....
Asymptotic result: log(a(n)) ~ 2*sqrt(Pi^2/6)*sqrt(n) as n -> inf. (End)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (12*Pi*sqrt(2*n)) * (1 - 73*Pi/(24*sqrt(6*n)) + (73/48 + 3601*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 05 2016
MAPLE
b:= proc(n, i) option remember; local g;
if n=0 or i=1 then [1, 0]
else g:= `if`(i>n, [0$2], b(n-i, i));
b(n, i-1) +g +[0, `if`(i=6, g[1], 0)]
fi
end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..100); # Alois P. Heinz, Oct 27 2012
MATHEMATICA
Table[ Count[ Flatten[ IntegerPartitions[n]], 6], {n, 1, 52} ]
b[n_, i_] := b[n, i] = Module[{g}, If [n == 0 || i == 1, {1, 0}, g = If[i > n, {0, 0}, b[n - i, i]]; b[n, i - 1] + g + {0, If[i == 6, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 09 2015, after Alois P. Heinz *)
CROSSREFS
Cf. A000041, A066633, A024786, A024787, A024788, A024789, A024791, A024792, A024793, A024794.
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling
STATUS
approved
A024794 Number of 10's in all partitions of n. +10
13
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 43, 57, 79, 104, 140, 183, 242, 312, 407, 520, 670, 849, 1081, 1359, 1715, 2141, 2678, 3322, 4125, 5085, 6274, 7691, 9430, 11502, 14025, 17024, 20655, 24959, 30140, 36270, 43612, 52274, 62604, 74763 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,12
COMMENTS
The sums of ten successive terms give A000070. - Omar E. Pol, Jul 12 2012
a(n) is also the difference between the sum of 10th largest and the sum of 11th largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012
In general, if m>0 and a(n+m)-a(n) = A000041(n), then a(n) ~ exp(sqrt(2*n/3)*Pi) / (2*Pi*m*sqrt(2*n)) * (1 - Pi*(1/24 + m/2)/sqrt(6*n) + (1/48 + Pi^2/6912 + m/4 + m*Pi^2/288 + m^2*Pi^2/72)/n). - Vaclav Kotesovec, Nov 05 2016
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.
Joseph Vandehey, Digital problems in the theory of partitions, Integers (2024) Vol. 24A, Art. No. A18. See p. 3.
FORMULA
a(n) = A181187(n,10) - A181187(n,11). - Omar E. Pol, Oct 25 2012
From Peter Bala, Dec 26 2013: (Start)
a(n+10) - a(n) = A000041(n). a(n) + a(n+5) = A024789(n).
a(n) + a(n+2) + a(n+4) + a(n+6) + a(n+8) = A024786(n).
O.g.f.: x^10/(1 - x^10) * product {k >= 1} 1/(1 - x^k) = x^10 + x^11 + 2*x^12 + 3*x^13 + ....
Asymptotic result: log(a(n)) ~ 2*sqrt(Pi^2/6)*sqrt(n) as n -> inf. (End)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (20*Pi*sqrt(2*n)) * (1 - 121*Pi/(24*sqrt(6*n)) + (121/48 + 9841*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 05 2016
MAPLE
b:= proc(n, i) option remember; local g;
if n=0 or i=1 then [1, 0]
else g:= `if`(i>n, [0$2], b(n-i, i));
b(n, i-1) +g +[0, `if`(i=10, g[1], 0)]
fi
end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..100); # Alois P. Heinz, Oct 27 2012
MATHEMATICA
Table[ Count[ Flatten[ IntegerPartitions[n]], 10], {n, 1, 55} ]
b[n_, i_] := b[n, i] = Module[{g}, If[n == 0 || i == 1, {1, 0}, g = If[i > n, {0, 0}, b[n - i, i]]; b[n, i - 1] + g + {0, If[i == 10, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 09 2015, after Alois P. Heinz *)
CROSSREFS
Cf. A066633, A000070(n-1), A024786, A024787, A024788, A024789, A024790, A024791, A024792, A024793, A000041.
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling
STATUS
approved
A024791 Number of 7's in all partitions of n. +10
12
0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 16, 23, 32, 45, 61, 84, 112, 151, 199, 263, 342, 446, 574, 739, 943, 1201, 1518, 1917, 2404, 3010, 3749, 4661, 5766, 7122, 8759, 10753, 13153, 16059, 19544, 23743, 28759, 34774, 41938, 50491, 60642, 72718, 87004, 103934, 123908 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,9
COMMENTS
The sums of seven successive terms give A000070. - Omar E. Pol, Jul 12 2012
a(n) is also the difference between the sum of 7th largest and the sum of 8th largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.
FORMULA
a(n) = A181187(n,7) - A181187(n,8). - Omar E. Pol, Oct 25 2012
a(n) ~ exp(Pi*sqrt(2*n/3)) / (14*Pi*sqrt(2*n)) * (1 - 85*Pi/(24*sqrt(6*n)) + (85/48 + 4873*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 05 2016
G.f.: x^7/(1 - x^7) * Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Apr 06 2017
MAPLE
b:= proc(n, i) option remember; local g;
if n=0 or i=1 then [1, 0]
else g:= `if`(i>n, [0$2], b(n-i, i));
b(n, i-1) +g +[0, `if`(i=7, g[1], 0)]
fi
end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..100); # Alois P. Heinz, Oct 27 2012
MATHEMATICA
<< DiscreteMath`Combinatorica`; Table[ Count[ Flatten[ Partitions[n]], 7], {n, 1, 52} ]
Table[Count[Flatten[IntegerPartitions[n]], 7], {n, 55}] (* Harvey P. Dale, Feb 26 2015 *)
b[n_, i_] := b[n, i] = Module[{g}, If[n == 0 || i == 1, {1, 0}, g = If[i > n, {0, 0}, b[n - i, i]]; b[n, i - 1] + g + {0, If[i == 7, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 09 2015, after Alois P. Heinz *)
PROG
(PARI) x='x+O('x^50); concat([0, 0, 0, 0, 0, 0], Vec(x^7/(1 - x^7) * prod(k=1, 50, 1/(1 - x^k)))) \\ Indranil Ghosh, Apr 06 2017
CROSSREFS
Cf. A066633, A024786, A024787, A024788, A024789, A024790, A024792, A024793, A024794.
KEYWORD
nonn
AUTHOR
Clark Kimberling
STATUS
approved
A024792 Number of 8's in all partitions of n. +10
12
0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 23, 31, 44, 59, 82, 108, 146, 191, 254, 328, 429, 549, 709, 900, 1148, 1446, 1829, 2286, 2865, 3559, 4427, 5465, 6752, 8288, 10178, 12429, 15175, 18442, 22404, 27102, 32767, 39473, 47516, 57012, 68349, 81703, 97579, 116236 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,10
COMMENTS
The sums of eight successive terms give A000070. - Omar E. Pol, Jul 12 2012
a(n) is also the difference between the sum of 8th largest and the sum of 9th largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.
FORMULA
a(n) = A181187(n,8) - A181187(n,9). - Omar E. Pol, Oct 25 2012
From Peter Bala, Dec 26 2013: (Start)
a(n+8) - a(n) = A000041(n). a(n) + a(n+4) = A024788(n).
a(n) + a(n+2) + a(n+4) + a(n+6) = A024786(n).
O.g.f.: x^8/(1 - x^8) * product {k >= 1} 1/(1 - x^k) = x^8 + x^9 + 2*x^10 + 3*x^11 + ....
Asymptotic result: log(a(n)) ~ 2*sqrt(Pi^2/6)*sqrt(n) as n -> inf. (End)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (16*Pi*sqrt(2*n)) * (1 - 97*Pi/(24*sqrt(6*n)) + (97/48 + 6337*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 05 2016
MAPLE
b:= proc(n, i) option remember; local g;
if n=0 or i=1 then [1, 0]
else g:= `if`(i>n, [0$2], b(n-i, i));
b(n, i-1) +g +[0, `if`(i=8, g[1], 0)]
fi
end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..100); # Alois P. Heinz, Oct 27 2012
MATHEMATICA
Table[ Count[ Flatten[ IntegerPartitions[n]], 8], {n, 1, 53} ]
(* second program: *)
b[n_, i_] := b[n, i] = Module[{g}, If[n == 0 || i == 1, {1, 0}, g = If[i > n, {0, 0}, b[n - i, i]]; b[n, i - 1] + g + {0, If[i == 8, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 09 2015, after Alois P. Heinz *)
CROSSREFS
Cf. A000041, A066633, A024786, A024787, A024788, A024789, A024790, A024791, A024793, A024794.
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling
STATUS
approved
A024793 Number of 9's in all partitions of n. +10
12
0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 31, 43, 58, 80, 106, 142, 187, 246, 319, 416, 533, 685, 872, 1108, 1397, 1762, 2204, 2755, 3426, 4251, 5250, 6476, 7950, 9746, 11905, 14514, 17638, 21403, 25888, 31265, 37661, 45288, 54329, 65079, 77775 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,11
COMMENTS
The sums of nine successive terms give A000070. - Omar E. Pol, Jul 12 2012
a(n) is also the difference between the sum of 9th largest and the sum of 10th largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.
FORMULA
a(n) = A181187(n,9) - A181187(n,10). - Omar E. Pol, Oct 25 2012
From Peter Bala, Dec 26 2013: (Start)
a(n+9) - a(n) = A000041(n).
a(n) + a(n+3) + a(n+6) = A024787(n).
O.g.f.: x^9/(1 - x^9) * product {k >= 1} 1/(1 - x^k) = x^9 + x^10 + 2*x^11 + 3*x^12 + ....
Asymptotic result: log(a(n)) ~ 2*sqrt(Pi^2/6)*sqrt(n) as n -> inf. (End)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (18*Pi*sqrt(2*n)) * (1 - 109*Pi/(24*sqrt(6*n)) + (109/48 + 7993*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 05 2016
MAPLE
b:= proc(n, i) option remember; local g;
if n=0 or i=1 then [1, 0]
else g:= `if`(i>n, [0$2], b(n-i, i));
b(n, i-1) +g +[0, `if`(i=9, g[1], 0)]
fi
end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..100); # Alois P. Heinz, Oct 27 2012
MATHEMATICA
Table[ Count[ Flatten[ IntegerPartitions[n]], 9], {n, 1, 55} ]
(* second program: *)
b[n_, i_] := b[n, i] = Module[{g}, If[n == 0 || i == 1, {1, 0}, g = If[i > n, {0, 0}, b[n - i, i]]; b[n, i - 1] + g + {0, If[i == 9, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 09 2015, after Alois P. Heinz *)
CROSSREFS
Cf. A000041, A066633, A024786, A024787, A024788, A024789, A024790, A024791, A024792, A024794.
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling
STATUS
approved
A182713 Number of 3's in the last section of the set of partitions of n. +10
7
0, 0, 1, 0, 1, 2, 2, 3, 6, 6, 10, 14, 18, 24, 35, 42, 58, 76, 97, 124, 164, 202, 261, 329, 412, 514, 649, 795, 992, 1223, 1503, 1839, 2262, 2741, 3346, 4056, 4908, 5919, 7150, 8568, 10297, 12320, 14721, 17542, 20911, 24808, 29456, 34870, 41232, 48652, 57389 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
COMMENTS
Also number of 3's in all partitions of n that do not contain 1 as a part.
Also 0 together with the first differences of A024787. - Omar E. Pol, Nov 13 2011
a(n) is the number of partitions of n having fewer 1s than 2s; e.g., a(7) counts these 3 partitions: [5, 2], [3, 2, 2], [2, 2, 2, 1]. - Clark Kimberling, Mar 31 2014
The last section of the set of partitions of n is also the n-th section of the set of partitions of any integer >= n. - Omar E. Pol, Apr 07 2014
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
FORMULA
It appears that A000041(n) = a(n+1) + a(n+2) + a(n+3), n >= 0. - Omar E. Pol, Feb 04 2012
a(n) ~ A000041(n)/3 ~ exp(Pi*sqrt(2*n/3)) / (12*sqrt(3)*n). - Vaclav Kotesovec, Jan 03 2019
EXAMPLE
a(7) = 2 counts the 3's in 7 = 4+3 = 3+2+2. The 3's in 7 = 3+3+1 = 3+2+1+1 = 3+1+1+1+1 do not count.
From Omar E. Pol, Oct 27 2012: (Start)
--------------------------------------
Last section Number
of the set of of
partitions of 7 3's
--------------------------------------
7 .............................. 0
4 + 3 .......................... 1
5 + 2 .......................... 0
3 + 2 + 2 ...................... 1
. 1 .......................... 0
. 1 ...................... 0
. 1 ...................... 0
. 1 .................. 0
. 1 ...................... 0
. 1 .................. 0
. 1 .................. 0
. 1 .............. 0
. 1 .............. 0
. 1 .......... 0
. 1 ...... 0
------------------------------------
. 5 - 3 = 2
.
In the last section of the set of partitions of 7 the difference between the sum of the third column and the sum of the fourth column is 5 - 3 = 2 equaling the number of 3's, so a(7) = 2 (see also A024787).
(End)
MAPLE
b:= proc(n, i) option remember; local g, h;
if n=0 then [1, 0]
elif i<2 then [0, 0]
else g:= b(n, i-1); h:= `if`(i>n, [0, 0], b(n-i, i));
[g[1]+h[1], g[2]+h[2]+`if`(i=3, h[1], 0)]
fi
end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..70); # Alois P. Heinz, Mar 18 2012
MATHEMATICA
z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Count[p, 1] < Count[p, 2]], {n, 0, z}] (* Clark Kimberling, Mar 31 2014 *)
b[n_, i_] := b[n, i] = Module[{g, h}, If[n == 0, {1, 0}, If[i<2, {0, 0}, g = b[n, i-1]; h = If[i>n, {0, 0}, b[n-i, i]]; Join[g[[1]] + h[[1]], g[[2]] + h[[2]] + If[i == 3, h[[1]], 0]]]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Nov 30 2015, after Alois P. Heinz *)
Table[Count[Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], 3], {n, 51}] (* Robert Price, May 15 2020 *)
PROG
(Sage) A182713 = lambda n: sum(list(p).count(3) for p in Partitions(n) if 1 not in p) # D. S. McNeil, Nov 29 2010
CROSSREFS
Column 3 of A194812.
Cf. A135010, A138121, A174455, A182703, A182712, A182714, A240056.
KEYWORD
nonn
AUTHOR
Omar E. Pol, Nov 28 2010
STATUS
approved
A173239 Triangle by columns, A000041 shifted down thrice, k>=0. +10
4
1, 1, 2, 3, 1, 5, 1, 7, 2, 11, 3, 1, 15, 5, 1, 22, 7, 2, 30, 11, 3, 1, 42, 15, 5, 1, 56, 22, 7, 2, 77, 30, 11, 3, 1, 101, 42, 15, 5, 1, 135, 56, 22, 7, 2, 176, 77, 30, 11, 3, 1, 231, 101, 42, 15, 5, 1, 297, 135, 56, 22, 7, 2, 385, 176, 77, 30, 11, 3, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Row sums = A024787, the numbers of 3's in all partitions of n, where A024787 starts with offset 1: (0, 0, 1, 1, 2, 4, 6, 9, 15,...). Triangle A173239 row sums start with the first "1" of A024787.
Let the triangle = M as an infinite lower triangular matrix. Then Lim_{n->inf} = A173241, the Euler transform of A051064 (the ruler function for 3).
Let P(x) = polcoeff A000041 = (1 + x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + ...), then P(x) = A(x) / A(x^3), where A(x) = polcoeff A173241: (1 + x + 2x^2 + 4x^3 + 6x^4 + ...)
Refer to A173238 comments for three conjectures relating A000041 to the infinite set of generalized ruler function sequences.
LINKS
Table of n, a(n) for n=0..69.
FORMULA
T(n,k) = A000041(n-3*k) for k=0..floor(n/3).
EXAMPLE
First few rows of the triangle =
1;
1;
2;
3, 1;
5, 1;
7, 2;
11, 3, 1;
15, 5, 1;
22, 7, 2;
30, 11, 3, 1;
42, 15, 5, 1;
56, 22, 7, 2;
77, 30, 11, 3, 1;
101, 42, 15, 5, 1;
135, 56, 22, 7, 2;
176, 77, 30, 11, 3, 1;
231, 101, 42, 15, 5, 1;
297, 135, 56, 22, 7, 2;
385, 176, 77, 30, 11, 3, 1;
490, 231, 101, 42, 15, 5, 1;
627, 297, 135, 56, 22, 7, 2;
792, 385, 176, 77, 30, 11, 3, 1;
1002,490, 231, 101, 42, 15, 5, 1;
1255, 627, 297, 135, 56, 22, 7, 2;
1575, 792, 385, 176, 77, 30, 11, 3, 1;
...
CROSSREFS
Cf. A000041, A173238, A173241, A051064, A024787.
KEYWORD
nonn,tabf,easy
AUTHOR
Gary W. Adamson, Feb 13 2010
STATUS
approved
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