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a(n) is the minimal position at which the maximal value of row n appears in row n of triangle A008289.
+20
3
1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7
COMMENTS
Except rows 3, 4, 10, 11, 21 of triangle A008289, all the other rows up to row number 10^6 contain a single maximal value. Conjecture: for n >= 22 there is a unique maximal value in row n of triangle A008289.
FORMULA
a(n) = min argmax(k->Q(n,k), k=1..m), that is a(n) = min{k, Q(n,k) = max{Q(n,p), p=1..m}}, where m = A003056(n) and Q(n,k) is defined by A008289. a(n) ~ K*sqrt(n) + O(1), where K = 2*sqrt(3)*log(2)/Pi = 0.76430413884... ( A131266).
EXAMPLE
For n=9, a(9)=2 because A003056(9)=3 and max{Q(9,p), p=1..3}=4 and Q(9,2)=4.
PROG
(PARI)
Q(N) = {
my(q = vector(N)); q[1] = [1, 0, 0, 0];
for (n = 2, N,
my(m = (sqrtint(8*n+1) - 1)\2);
q[n] = vector((1 + (m>>2)) << 2); q[n][1] = 1;
for (k = 2, m, q[n][k] = q[n-k][k] + q[n-k][k-1]));
return(q);
};
seq(N) = {
my(a = vector(N), q = Q(N), vmx = apply(vecmax, q));
for (n = 1, N, a[n] = vecmin(select(v->v==vmx[n], q[n], 1)));
a;
};
Eigenvector of the triangle of distinct partitions ( A008289), so that: a(n) = Sum_{k=1..tri(n)} A008289(n,k)*a(k) for n>=1 with a(1)=1, where tri(n) = floor((sqrt(8*n+1)-1)/2).
+20
2
1, 1, 2, 2, 3, 5, 6, 8, 11, 15, 18, 24, 29, 37, 47, 57, 69, 86, 103, 125, 154, 183, 220, 264, 316, 375, 450, 533, 631, 747, 882, 1035, 1222, 1428, 1674, 1959, 2282, 2653, 3088, 3578, 4142, 4790, 5525, 6363, 7330, 8410, 9644, 11050, 12633, 14424, 16459, 18743
MAPLE
b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
-> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))
end:
a:= proc(n) option remember; local l; l:= b(n, n);
`if`(n=1, 1, add(l[i+1]*a(i), i=1..nops(l)-1))
end:
MATHEMATICA
zip = With[{m = Max[Length[#1], Length[#2]]}, PadRight[#1, m] + PadRight[#2, m]]&; b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i < 1, {}, zip[b[n, i-1], If[i>n, {}, Join[{0}, b[n-i, i-1]]]]]]; a[n_] := a[n] = ( l = b[n, n]; If[n == 1, 1, Sum[l[[i+1]]*a[i], {i, 1, Length[l]-1}]]); Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Feb 12 2017, after Alois P. Heinz *)
PROG
(PARI) {a(n)=if(n<0, 0, if(n==1, 1, sum(k=1, floor((sqrt(8*n+1)-1)/2), a(k)*polcoeff(polcoeff(prod(i=1, n, 1+y*x^i, 1+x*O(x^n)), n, x), k, y))))}
1, 1, 1, 2, 2, 5, 6, 11, 14, 26, 34, 52, 79, 110, 162, 231, 328, 446, 660, 872, 1236, 1660, 2327, 3060, 4238, 5586, 7595, 9988, 13376, 17470, 23318, 30286, 39744, 51676, 67276, 86929, 112390, 144472, 185428, 237826, 303171, 386468, 490838, 622882, 786068, 995118, 1249838, 1573522, 1970542
FORMULA
a(n) = sum(q(n,k)^k,k=0..n), where q(n,k) is the number of partitions of n into k distinct parts ( A008289, A060016).
MATHEMATICA
qq[n_] := CoefficientList[Series[Sum[x^Binomial[k+1, 2]y^k/Product[1-x^i, {i, 1, k}], {k, 0, n}], {x, 0, n}, {y, 0, n}], {x, y}]
Total[Transpose[Map[#^2 &, qq[100]]]]
Number of compositions (ordered partitions) of n into distinct parts.
+10
216
1, 1, 1, 3, 3, 5, 11, 13, 19, 27, 57, 65, 101, 133, 193, 351, 435, 617, 851, 1177, 1555, 2751, 3297, 4757, 6293, 8761, 11305, 15603, 24315, 30461, 41867, 55741, 74875, 98043, 130809, 168425, 257405, 315973, 431065, 558327, 751491, 958265, 1277867, 1621273
COMMENTS
a(n) = the number of different ways to run up a staircase with n steps, taking steps of distinct sizes where the order matters and there is no other restriction on the number or the size of each step taken. - Mohammad K. Azarian, May 21 2008 Compositions into distinct parts are equivalent to (1,1)-avoiding compositions. - Gus Wiseman, Jun 25 2020
REFERENCES
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17.
FORMULA
"AGK" (ordered, elements, unlabeled) transform of 1, 1, 1, 1, ...
G.f.: Sum_{k>=0} k! * x^((k^2+k)/2) / Product_{j=1..k} (1-x^j). - David W. Wilson May 04 2000
EXAMPLE
a(6) = 11 because 6 = 5+1 = 4+2 = 3+2+1 = 3+1+2 = 2+4 = 2+3+1 = 2+1+3 = 1+5 = 1+3+2 = 1+2+3.
The a(0) = 1 through a(7) = 13 strict compositions:
() (1) (2) (3) (4) (5) (6) (7)
(1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (3,1) (2,3) (2,4) (2,5)
(3,2) (4,2) (3,4)
(4,1) (5,1) (4,3)
(1,2,3) (5,2)
(1,3,2) (6,1)
(2,1,3) (1,2,4)
(2,3,1) (1,4,2)
(3,1,2) (2,1,4)
(3,2,1) (2,4,1)
(4,1,2)
(4,2,1)
(End)
MAPLE
b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
-> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0))) end:
a:= proc(n) local l; l:=b(n, n): add((i-1)! *l[i], i=1..nops(l)) end:
# second Maple program:
T:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
`if`(k=0, `if`(n=0, 1, 0), T(n-k, k) +k*T(n-k, k-1)))
end:
a:= n-> add(T(n, k), k=0..floor((sqrt(8*n+1)-1)/2)):
MATHEMATICA
f[list_]:=Length[list]!; Table[Total[Map[f, Select[IntegerPartitions[n], Sort[#] == Union[#] &]]], {n, 0, 30}]
T[n_, k_] := T[n, k] = If[k<0 || n<0, 0, If[k==0, If[n==0, 1, 0], T[n-k, k] + k*T[n-k, k-1]]]; a[n_] := Sum[T[n, k], {k, 0, Floor[(Sqrt[8*n + 1] - 1) / 2]}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Sep 22 2015, after Alois P. Heinz *)
PROG
(PARI)
N=66; q='q+O('q^N);
gf=sum(n=0, N, n!*q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) );
Vec(gf)
(PARI)
my(q = vector(N)); q[1] = [1, 0, 0, 0];
for (n = 2, N,
my(m = (sqrtint(8*n+1) - 1)\2);
q[n] = vector((1 + (m>>2)) << 2); q[n][1] = 1;
for (k = 2, m, q[n][k] = q[n-k][k] + q[n-k][k-1]));
return(q);
};
seq(N) = concat(1, apply(q -> sum(k = 1, #q, q[k] * k!), Q(N)));
CROSSREFS
Dominated by A003242 (anti-run compositions). These compositions are ranked by A233564. (1,1)-avoiding patterns are counted by A000142. Numbers with strict prime signature are A130091. (1,1,1)-avoiding compositions are counted by A232432. (1,1)-matching compositions are counted by A261982. Inseparable partitions are counted by A325535. Patterns matched by compositions are counted by A335456. Strict permutations of prime indices are counted by A335489.
Sphenic numbers: products of 3 distinct primes.
(Formerly M5207)
+10
193
30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 345, 354, 357, 366, 370, 374, 385, 399, 402, 406, 410, 418, 426, 429, 430, 434, 435, 438
COMMENTS
Note the distinctions between this and "n has exactly three prime factors" ( A014612) or "n has exactly three distinct prime factors." ( A033992). The word "sphenic" also means "shaped like a wedge" [American Heritage Dictionary] as in dentation with "sphenic molars." - Jonathan Vos Post, Sep 11 2005 Also the volume of a sphenic brick. A sphenic brick is a rectangular parallelepiped whose sides are components of a sphenic number, namely whose sides are three distinct primes. Example: The distinct prime triple (3,5,7) produces a 3x5x7 unit brick which has volume 105 cubic units. 3-D analog of 2-D A037074 Product of twin primes, per Cino Hilliard's comment. Compare with 3-D A107768 Golden 3-almost primes = Volumes of bricks (rectangular parallelepipeds) each of whose faces has golden semiprime area. - Jonathan Vos Post, Jan 08 2007 Sum(n>=1, 1/a(n)^s) = (1/6)*(P(s)^3 - P(3*s) - 3*(P(s)*P(2*s)-P(3*s))), where P is prime zeta function. - Enrique Pérez Herrero, Jun 28 2012 n = 265550 is the smallest n with a(n) (=1279789) < A006881(n) (=1279793). - Peter Dolland, Apr 11 2020
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
"Sphenic", The American Heritage Dictionary of the English Language, Fourth Edition, Houghton Mifflin Company, 2000.
EXAMPLE
Also Heinz numbers of strict integer partitions into three parts, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). These partitions are counted by A001399(n-6) = A069905(n-3), with ordered version A001399(n-6)*6. The sequence of terms together with their prime indices begins: 30: {1,2,3} 182: {1,4,6} 286: {1,5,6}
42: {1,2,4} 186: {1,2,11} 290: {1,3,10}
66: {1,2,5} 190: {1,3,8} 310: {1,3,11}
70: {1,3,4} 195: {2,3,6} 318: {1,2,16}
78: {1,2,6} 222: {1,2,12} 322: {1,4,9}
102: {1,2,7} 230: {1,3,9} 345: {2,3,9}
105: {2,3,4} 231: {2,4,5} 354: {1,2,17}
110: {1,3,5} 238: {1,4,7} 357: {2,4,7}
114: {1,2,8} 246: {1,2,13} 366: {1,2,18}
130: {1,3,6} 255: {2,3,7} 370: {1,3,12}
138: {1,2,9} 258: {1,2,14} 374: {1,5,7}
154: {1,4,5} 266: {1,4,8} 385: {3,4,5}
165: {2,3,5} 273: {2,4,6} 399: {2,4,8}
170: {1,3,7} 282: {1,2,15} 402: {1,2,19}
174: {1,2,10} 285: {2,3,8} 406: {1,4,10}
(End)
MAPLE
with(numtheory): a:=proc(n) if bigomega(n)=3 and nops(factorset(n))=3 then n else fi end: seq(a(n), n=1..450); # Emeric Deutsch option remember;
local a;
if n =1 then
30;
else
for a from procname(n-1)+1 do
if bigomega(a)=3 and nops(factorset(a))=3 then
return a;
end if;
end do:
end if;
MATHEMATICA
Union[Flatten[Table[Prime[n]*Prime[m]*Prime[k], {k, 20}, {n, k+1, 20}, {m, n+1, 20}]]]
Take[ Sort@ Flatten@ Table[ Prime@i Prime@j Prime@k, {i, 3, 21}, {j, 2, i - 1}, {k, j - 1}], 53] (* Robert G. Wilson v *) With[{upto=500}, Sort[Select[Times@@@Subsets[Prime[Range[Ceiling[upto/6]]], {3}], #<=upto&]]] (* Harvey P. Dale, Jan 08 2015 *) Select[Range[100], SquareFreeQ[#]&&PrimeOmega[#]==3&] (* Gus Wiseman, Nov 05 2020 *)
PROG
(PARI) list(lim)=my(v=List(), t); forprime(p=2, (lim)^(1/3), forprime(q=p+1, sqrt(lim\p), t=p*q; forprime(r=q+1, lim\t, listput(v, t*r)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011 (Haskell)
a007304 n = a007304_list !! (n-1)
a007304_list = filter f [1..] where
f u = p < q && q < w && a010051 w == 1 where
p = a020639 u; v = div u p; q = a020639 v; w = div v q
(Python)
from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a, k in enumerate(primerange(integer_nthroot(x, 3)[0]+1), 1) for b, m in enumerate(primerange(k+1, isqrt(x//k)+1), a+1)))
kmin, kmax = 0, 1
while f(kmax) > kmax:
kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
CROSSREFS
Cf. A002033, A010051, A020639, A037074, A046393, A061299, A067467, A071140, A096917, A096918, A096919, A100765, A103653, A107464, A107768, A179643, A179695. For the following, NNS means "not necessarily strict".
A008289 counts strict partitions by sum and length.
A220377 counts 3-part pairwise coprime strict partitions (NNS: A307719).
EXTENSIONS
Comment concerning number of divisors corrected by R. J. Mathar, Aug 14 2009
Triangle read by rows where T(n,k) is the number of integer partitions of n with median k = 1..n.
+10
97
1, 1, 1, 1, 0, 1, 2, 2, 0, 1, 3, 1, 0, 0, 1, 4, 2, 3, 0, 0, 1, 6, 3, 1, 0, 0, 0, 1, 8, 6, 2, 4, 0, 0, 0, 1, 11, 7, 3, 1, 0, 0, 0, 0, 1, 15, 10, 4, 2, 5, 0, 0, 0, 0, 1, 20, 13, 7, 3, 1, 0, 0, 0, 0, 0, 1, 26, 19, 11, 4, 2, 6, 0, 0, 0, 0, 0, 1
COMMENTS
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
Triangle begins:
1
1 1
1 0 1
2 2 0 1
3 1 0 0 1
4 2 3 0 0 1
6 3 1 0 0 0 1
8 6 2 4 0 0 0 1
11 7 3 1 0 0 0 0 1
15 10 4 2 5 0 0 0 0 1
20 13 7 3 1 0 0 0 0 0 1
26 19 11 4 2 6 0 0 0 0 0 1
35 24 14 5 3 1 0 0 0 0 0 0 1
45 34 17 8 4 2 7 0 0 0 0 0 0 1
58 42 23 12 5 3 1 0 0 0 0 0 0 0 1
For example, row n = 9 counts the following partitions:
(7,1,1) (5,2,2) (3,3,3) (4,4,1) . . . . (9)
(6,1,1,1) (6,2,1) (4,3,2)
(3,3,1,1,1) (3,2,2,2) (5,3,1)
(4,2,1,1,1) (4,2,2,1)
(5,1,1,1,1) (4,3,1,1)
(3,2,1,1,1,1) (2,2,2,2,1)
(4,1,1,1,1,1) (3,2,2,1,1)
(2,2,1,1,1,1,1)
(3,1,1,1,1,1,1)
(2,1,1,1,1,1,1,1)
(1,1,1,1,1,1,1,1,1)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Median[#]==k&]], {n, 15}, {k, n}]
CROSSREFS
Including half-steps gives A359893. The median statistic is ranked by A360005(n)/2. A240219 counts partitions w/ the same mean as median, complement A359894.
Square array T(n,k) read by antidiagonals giving number of ways to distribute n indistinguishable objects in k indistinguishable containers; containers may be left empty.
+10
88
1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 3, 3, 2, 1, 1, 0, 1, 3, 4, 3, 2, 1, 1, 0, 1, 4, 5, 5, 3, 2, 1, 1, 0, 1, 4, 7, 6, 5, 3, 2, 1, 1, 0, 1, 5, 8, 9, 7, 5, 3, 2, 1, 1, 0, 1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1, 0, 1, 6, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1, 0, 1, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 1
COMMENTS
T(n,k) is also the number of partitions of n with greatest part k, if we assume the greatest part of an empty partition to be 0. Row n = 9 counts the following partitions:
111111111 22221 333 432 54 63 72 81 9
222111 3222 441 522 621 711
2211111 3321 4221 531 6111
21111111 32211 4311 5211
33111 42111 51111
321111 411111
3111111
(End)
FORMULA
T(0, k) = 1, T(n, 0) = 0 (n>0), T(1, k) = 1 (k>0), T(n, 1) = 1 (n>0), T(n, k) = 0 for n < 0, T(n, k) = Sum[ T(n-k+i, k-i), i=0...k-1] Or, T(n, 1) = T(n, n) = 1, T(n, k) = 0 (k>n), T(n, k) = T(n-1, k-1) + T(n-k, k).
G.f. Product_{j=0..infinity} 1/(1-xy^j). Regarded as a triangular array, g.f. Product_{j=1..infinity} 1/(1-xy^j). - Franklin T. Adams-Watters, Dec 18 2006 O.g.f. of column No. k of the triangle a(n,k) is x^k/product(1-x^j,j=1..k), k>=0 (the undefined product for k=0 is put to 1). - Wolfdieter Lang, Dec 03 2012
EXAMPLE
Table begins (upper left corner = T(0,0)):
1 1 1 1 1 1 1 1 1 ...
0 1 1 1 1 1 1 1 1 ...
0 1 2 2 2 2 2 2 2 ...
0 1 2 3 3 3 3 3 3 ...
0 1 3 4 5 5 5 5 5 ...
0 1 3 5 6 7 7 7 7 ...
0 1 4 7 9 10 11 11 11 ...
0 1 4 8 11 13 14 15 15 ...
0 1 5 10 15 18 20 21 22 ...
There is 1 way to distribute 0 objects into k containers: T(0, k) = 1. The different ways for n=4, k=3 are: (oooo)()(), (ooo)(o)(), (oo)(oo)(), (oo)(o)(o), so T(4, 3) = 4.
The triangle a(n,k) = T(n-k,k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 ...
00 1
01 0 1
02 0 1 1
03 0 1 1 1
04 0 1 2 1 1
05 0 1 2 2 1 1
06 0 1 3 3 2 1 1
07 0 1 3 4 3 2 1 1
08 0 1 4 5 5 3 2 1 1
09 0 1 4 7 6 5 3 2 1 1
10 0 1 5 8 9 7 5 3 2 1 1
...
Row n=5 is, for k=1..5, [1,2,2,1,1] which gives the number of partitions of n=5 with k parts. See A008284 and the Franklin T. Adams-Watters comment above. (End) Row n = 9 counts the following partitions:
9 54 333 3222 22221 222111 2211111 21111111 111111111
63 432 3321 32211 321111 3111111
72 441 4221 33111 411111
81 522 4311 42111
531 5211 51111
621 6111
711
(End)
MATHEMATICA
Flatten[Table[Length[IntegerPartitions[n, {k}]], {n, 0, 20}, {k, 0, n}]] (* Emanuele Munarini, Feb 24 2014 *)
PROG
(Sage)
from sage.combinat.partition import number_of_partitions_length
[[number_of_partitions_length(n, k) for k in (0..n)] for n in (0..10)] # Peter Luschny, Aug 01 2015
CROSSREFS
Sum of antidiagonal entries T(n, k) with n+k=m equals A000041(m). The version for factorizations is A316439.
AUTHOR
Martin Wohlgemuth (mail(AT)matroid.com), Jul 05 2002
Number of partitions of n into distinct parts such that number of parts is odd.
+10
85
0, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 19, 23, 27, 32, 38, 44, 52, 61, 71, 82, 96, 111, 128, 148, 170, 195, 224, 256, 293, 334, 380, 432, 491, 557, 630, 713, 805, 908, 1024, 1152, 1295, 1455, 1632, 1829, 2048, 2291, 2560, 2859, 3189, 3554, 3958, 4404
COMMENTS
Ramanujan theta functions: phi(q) := Sum_{k=-oo..oo} q^(k^2) ( A000122), chi(q) := Prod_{k>=0} (1+q^(2k+1)) ( A000700).
FORMULA
Expansion of (1-phi(-q))/(2*chi(-q)) in powers of q where phi(),chi() are Ramanujan theta functions. - Michael Somos, Feb 14 2006 G.f.: sum(n>=1, q^(2*n^2-n) / prod(k=1..2*n-1, 1-q^k ) ). [ Joerg Arndt, Apr 01 2014]
EXAMPLE
The a(5) = 1 through a(15) = 14 partitions (A-F = 10..15):
5 6 7 8 9 A B C D E F
321 421 431 432 532 542 543 643 653 654
521 531 541 632 642 652 743 753
621 631 641 651 742 752 762
721 731 732 751 761 843
821 741 832 842 852
831 841 851 861
921 931 932 942
A21 941 951
A31 A32
B21 A41
B31
C21
54321
(End)
MAPLE
b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
`if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
end:
a:= n-> b(n$2, 0):
MATHEMATICA
b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 1)/2, 0, If[n == 0, t, Sum[b[n - i*j, i - 1, Abs[t - j]], {j, 0, Min[n/i, 1]}]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jan 16 2015, after Alois P. Heinz *) CoefficientList[Normal[Series[(QPochhammer[-x, x]-QPochhammer[x])/2, {x, 0, 100}]], x] (* Andrey Zabolotskiy, Apr 12 2017 *) Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&OddQ[Length[#]]&]], {n, 0, 30}] (* Gus Wiseman, Jan 09 2021 *)
PROG
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)/eta(x+A) - eta(x+A))/2, n))} /* Michael Somos, Feb 14 2006 */ (PARI) N=66; q='q+O('q^N); S=1+2*sqrtint(N);
gf=sum(n=1, S, (n%2!=0) * q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) );
(PARI) N=66; q='q+O('q^N); S=1+sqrtint(N);
gf=sum(n=1, S, q^(2*n^2-n) / prod(k=1, 2*n-1, 1-q^k ) );
CROSSREFS
Numbers with these strict partitions as binary indices are A000069. The Heinz numbers of these partitions are A030059. Other cases of odd length:
- A024429 counts set partitions of odd length. - A089677 counts ordered set partitions of odd length. - A174726 counts ordered factorizations of odd length. - A339890 counts factorizations of odd length. A008289 counts strict partitions by sum and length.
A026804 counts partitions whose least part is odd, with strict case A026832.
Number of strict integer partitions of n such that no part can be written as a nonnegative linear combination of the others.
+10
81
1, 1, 1, 1, 1, 2, 1, 3, 2, 3, 3, 5, 3, 6, 5, 7, 6, 9, 7, 11, 10, 14, 12, 16, 15, 20, 17, 24, 22, 27, 29, 32, 30, 41, 36, 49, 45, 50, 52, 65, 63, 70, 77, 80, 83, 104, 98, 107, 116, 126, 134, 152, 148, 162, 180, 196, 195, 227, 227, 238, 272, 271, 293, 333, 325
COMMENTS
A way of writing n as a (presumed nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).
EXAMPLE
The a(16) = 6 through a(22) = 12 strict partitions:
(16) (17) (18) (19) (20) (21) (22)
(9,7) (9,8) (10,8) (10,9) (11,9) (12,9) (13,9)
(10,6) (10,7) (11,7) (11,8) (12,8) (13,8) (14,8)
(11,5) (11,6) (13,5) (12,7) (13,7) (15,6) (15,7)
(13,3) (12,5) (14,4) (13,6) (14,6) (16,5) (16,6)
(7,5,4) (13,4) (7,6,5) (14,5) (17,3) (17,4) (17,5)
(14,3) (8,7,3) (15,4) (8,7,5) (19,2) (18,4)
(15,2) (16,3) (9,6,5) (11,10) (19,3)
(7,6,4) (17,2) (9,7,4) (8,7,6) (12,10)
(8,6,5) (11,5,4) (9,7,5) (9,7,6)
(9,6,4) (10,7,4) (9,8,5)
(10,8,3) (7,6,5,4)
(11,6,4)
(11,7,3)
MATHEMATICA
combs[n_, y_]:=With[{s=Table[{k, i}, {k, y}, {i, 0, Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&And@@Table[combs[#[[k]], Delete[#, k]]=={}, {k, Length[#]}]&]], {n, 0, 15}]
PROG
(Python)
from sympy.utilities.iterables import partitions
if n <= 1: return 1
alist, c = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)], 1
for p in partitions(n, k=n-1):
if max(p.values(), default=0)==1:
s = set(p)
if not any(set(t).issubset(s-{q}) for q in s for t in alist[q]):
c += 1
CROSSREFS
For sums of subsets instead of combinations of partitions we have A151897. For subsets instead of partitions we have A326083, complement A364914. A more strict variation is A364915. The case of all positive coefficients is A365006. A364912 counts linear combinations of partitions of k.
Cf. A007865, A085489, A237113, A275972, A363226, A364272, A364533, A364910, A364911, A365002, A365004.
Number of strict integer partitions of n containing the sum of some subset of the parts. A variation of sum-full strict partitions.
+10
78
0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3, 1, 4, 3, 8, 6, 11, 10, 17, 16, 26, 25, 39, 39, 54, 60, 82, 84, 116, 126, 160, 177, 222, 242, 302, 337, 402, 453, 542, 601, 722, 803, 936, 1057, 1234, 1373, 1601, 1793, 2056, 2312, 2658, 2950, 3395, 3789, 4281, 4814, 5452, 6048
COMMENTS
First differs from A316402 at a(16) = 11 due to (7,5,3,1).
EXAMPLE
The a(6) = 1 through a(16) = 11 partitions (A=10):
(321) . (431) . (532) (5321) (642) (5431) (743) (6432) (853)
(541) (651) (6421) (752) (6531) (862)
(4321) (5421) (7321) (761) (7431) (871)
(6321) (5432) (7521) (6532)
(6431) (9321) (6541)
(6521) (54321) (7432)
(7421) (7621)
(8321) (8431)
(8521)
(A321)
(64321)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Intersection[#, Total/@Subsets[#, {2, Length[#]}]]!={}&]], {n, 0, 30}]
CROSSREFS
The linear combination-free version is A364350.
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