Displaying 1-10 of 272 results found.
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1, 2, 7, 18, 47, 110, 258, 568, 1237, 2600, 5380, 10870, 21652, 42350, 81778, 155676, 292964, 544846, 1003078, 1828128, 3301952, 5911740, 10499385, 18502582, 32371011, 56240816, 97073055, 166497412, 283870383, 481212656, 811287037, 1360575284, 2270274785, 3769835178, 6230705170, 10251665550, 16794445441
COMMENTS
Equals [1,2,3,...] * [1,0,4,0,10,0,20,...] * [1,0,0,6,0,0,21,...] * [1,0,0,0,8,0,0,0,36,...] * ... - Gary W. Adamson, Jul 06 2009 Number of pairs of planar partitions of u and v where u + v = n. - Joerg Arndt, Apr 22 2014
FORMULA
G.f.: 1 / prod(k>=1, (1-x^k)^k )^2. - Joerg Arndt, Apr 22 2014 a(n) ~ Zeta(3)^(2/9) * exp(1/6 + 3*n^(2/3)*(Zeta(3)/2)^(1/3)) / (A^2 * 2^(1/18) * sqrt(3*Pi) * n^(13/18)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Feb 27 2015 G.f.: exp(2*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 29 2018
MAPLE
a:= proc(n) option remember; `if`(n=0, 1, 2*add(
a(n-j)*numtheory[sigma][2](j), j=1..n)/n)
end:
MATHEMATICA
nn = 36; CoefficientList[Series[Product[1/(1 - x^i)^(2 i), {i, 1, nn}] , {x, 0, nn}], x] (* Geoffrey Critzer, Nov 29 2014 *)
PROG
(PARI) N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1-x^k)^k)^2) \\ Joerg Arndt, Apr 22 2014
1, 2, 5, 11, 24, 48, 96, 182, 342, 624, 1124, 1983, 3462, 5947, 10114, 16993, 28290, 46624, 76225, 123555, 198833, 317627, 504102, 794885, 1246079, 1942112, 3010857, 4643515, 7126749, 10886361, 16555324, 25067633, 37801062, 56776035, 84951990, 126643036, 188127997, 278507781, 410949776, 604437277, 886284200, 1295668181
FORMULA
Euler transform of 2, 2, 3, 4, 5, 6, 7, 8, 9, ...
G.f.: 1/( (1-x) * prod(n>=1, (1-x^n)^n ) ). [ Joerg Arndt, Mar 15 2014] a(n) ~ (n/(2*Zeta(3)))^(1/3) * A000219(n). a(n) ~ exp(1/12 + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A * 2^(23/36) * sqrt(3*Pi) * Zeta(3)^(5/36) * n^(13/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. (End)
G.f.: exp(Sum_{k>=1} (sigma_2(k) + 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018
MATHEMATICA
CoefficientList[Series[1/(1-x)*Product[1/(1-x^k)^k, {k, 1, 50}], {x, 0, 50}], x] (* Vaclav Kotesovec, Aug 16 2015 *)
PROG
(PARI) N=66; x='x+O('x^N); Vec( 1/((1-x)*prod(n=1, N, (1-x^n)^n )) ) \\ Joerg Arndt, Mar 15 2014
Number of symmetrical planar partitions of n: planar partitions ( A000219) that when regarded as 3-D objects have a threefold axis of symmetry that is the intersection of 3 mirror planes, i.e., C3v symmetry.
+20
13
1, 0, 0, 1, 0, 0, 2, 1, 0, 2, 1, 0, 2, 1, 0, 3, 2, 0, 4, 4, 0, 4, 5, 0, 5, 7, 1, 6, 9, 1, 6, 11, 1, 8, 15, 2, 10, 20, 3, 10, 25, 4, 12, 33, 7, 14, 40, 9, 15, 48, 12, 18, 60, 17, 20, 74, 23, 22, 89, 30, 26, 108, 40, 30, 130, 51, 33, 157, 66, 37, 187, 85, 42, 222, 108, 47, 262, 136, 54
EXAMPLE
The plane partition {{2,1},{1}} has C3v symmetry.
Number of symmetrical planar partitions of n: planar partitions ( A000219) that when regarded as 3-D objects have only a threefold axis of symmetry, i.e., C3 symmetry.
+20
13
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 2, 2, 0, 3, 3, 0, 5, 6, 0, 7, 9, 0, 11, 16, 1, 14, 23, 2, 20, 36, 4, 27, 52, 7, 37, 78, 13, 48, 111, 21, 65, 163, 36, 83, 227, 56, 109, 322, 89, 139, 444, 135, 179, 618, 207, 226, 841, 305, 288, 1151, 453, 361
EXAMPLE
The plane partitions {{3, 2, 2}, {3, 1}, {1, 1}} and {{3, 2, 2}, {3, 2}, {1, 1}} have C3 symmetry.
0, 2, 3, 7, 11, 24, 38, 74, 122, 218, 359, 620, 1006, 1682, 2712, 4418, 7037, 11267, 17729, 27948, 43516, 67681, 104308, 160411, 244839, 372712, 563913, 850576, 1276378, 1909351, 2843346, 4221120, 6241544, 9200982, 13515091, 19793915, 28894823, 42062211, 61045506, 88359422, 127537058, 183617286, 263666228, 377696338, 539715276, 769456793
REFERENCES
G. Almkvist, The differences of the number of plane partitions, Manuscript, circa 1991.
FORMULA
a(n) ~ 2^(1/36) * Zeta(3)^(19/36) * exp(1/12 + 3*Zeta(3)^(1/3)*n^(2/3)/2^(2/3)) / (A * sqrt(3*Pi) * n^(37/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Oct 05 2015
MATHEMATICA
nmax = 60; Rest[CoefficientList[Series[Product[1/(1 - x^k)^k, {k, 2, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Oct 05 2015 *)
Number of asymmetrical planar partitions of n: planar partitions ( A000219) that when regarded as 3-D objects have no symmetry.
(Formerly M1392 N0542)
+20
10
0, 0, 0, 1, 2, 5, 11, 21, 39, 73, 129, 226, 388, 659, 1100, 1821, 2976, 4828, 7754, 12370, 19574, 30789, 48097, 74725, 115410, 177366, 271159, 412665, 625098, 942932, 1416362, 2119282, 3158840, 4691431, 6942882, 10240503, 15054705
REFERENCES
P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
MATHEMATICA
nmax = 150;
a219[0] = 1;
a219[n_] := a219[n] = Sum[a219[n - j] DivisorSigma[2, j], {j, n}]/n;
s = Product[1/(1 - x^(2 i - 1))/(1 - x^(2 i))^Floor[i/2], {i, 1, Ceiling[( nmax + 1)/2]}] + O[x]^( nmax + 1);
a048140[n_] := (a219[n] + A005987[[n + 1]])/2; A048141 = Cases[Import["https://oeis.org/ A048141/b048141.txt", "Table"], {_, _}][[All, 2]];
A048142 = Cases[Import["https://oeis.org/ A048142/b048142.txt", "Table"], {_, _}][[All, 2]];
a[1] = 0;
Binomial transform of the number of planar partitions ( A000219).
+20
10
1, 2, 6, 19, 60, 185, 559, 1662, 4875, 14134, 40564, 115370, 325465, 911355, 2534595, 7004827, 19246626, 52596377, 143006632, 386984573, 1042537831, 2796803110, 7473161196, 19893461042, 52767059608, 139488323734, 367540167625, 965445514862, 2528516552660
COMMENTS
Let 0 < p < 1, r > 0, v > 0, f(n) = v*exp(r*n^p)/n^b, then
Sum_{k=0..n} binomial(n,k) * f(k) ~ f(n/2) * 2^n * exp(g(n)), where
g(n) = p^2 * r^2 * n^p / (2^(1+2*p)*n^(1-p) + p*r*(1-p)*2^(1+p)).
Special cases:
p < 1/2, g(n) = 0
p = 1/2, g(n) = r^2/16
p = 2/3, g(n) = r^2 * n^(1/3) / (9 * 2^(1/3)) - r^3/81
p = 3/4, g(n) = 9*r^2*sqrt(n)/(64*sqrt(2)) - 27*r^3*n^(1/4)/(2048*2^(1/4)) + 81*r^4/65536
p = 3/5, g(n) = 9*r^2*n^(1/5)/(100*2^(1/5))
p = 4/5, g(n) = 2^(7/5)*r^2*n^(3/5)/25 - 4*2^(3/5)*r^3*n^(2/5)/625 + 8*2^(4/5)*r^4*n^(1/5)/15625 - 32*r^5/390625
FORMULA
a(n) = Sum_{k=0..n} binomial(n,k) * A000219(k). a(n) ~ exp(1/12 + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(4/3) + Zeta(3)^(2/3) * n^(1/3) / 2^(5/3) - Zeta(3)/12) * 2^(n + 7/18) * Zeta(3)^(7/36) / (A * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962. G.f.: (1/(1 - x))*exp(Sum_{k>=1} sigma_2(k)*x^k/(k*(1 - x)^k)). - Ilya Gutkovskiy, Aug 20 2018
MATHEMATICA
nmax = 40; s = CoefficientList[Series[Product[1/(1-x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]; Table[Sum[Binomial[n, k] * s[[k+1]], {k, 0, n}], {n, 0, nmax}]
Number of symmetrical planar partitions of n (planar partitions ( A000219) that when regarded as 3-D objects have just one symmetry plane).
(Formerly M0322 N0119)
+20
9
0, 1, 2, 2, 4, 6, 6, 11, 16, 20, 28, 41, 51, 70, 93, 122, 158, 211, 266, 350, 450, 577, 730, 948, 1186, 1510, 1901, 2408, 2999, 3790, 4703, 5898, 7310, 9111, 11231, 13979, 17168, 21229, 26036, 32095, 39188, 48155, 58657, 71798, 87262, 106472, 129014
REFERENCES
P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
MATHEMATICA
nmax = 150;
a219[0] = 1;
a219[n_] := a219[n] = Sum[a219[n - j] DivisorSigma[2, j], {j, n}]/n;
s = Product[1/(1 - x^(2 i - 1))/(1 - x^(2 i))^Floor[i/2], {i, 1, Ceiling[( nmax + 1)/2]}] + O[x]^( nmax + 1);
a048140[n_] := (a219[n] + A005987[[n + 1]])/2; A048141 = Cases[Import["https://oeis.org/ A048141/b048141.txt", "Table"], {_, _}][[All, 2]];
a[1] = 0;
a[n_] := - A048141[[n]] + 2 a048140[n] - a219[n];
Inverse binomial transform of the number of planar partitions ( A000219).
+20
5
1, 0, 2, -1, 4, -7, 19, -48, 123, -304, 728, -1694, 3865, -8735, 19739, -44875, 102818, -236939, 546988, -1260023, 2888607, -6584008, 14927816, -33714166, 75976024, -171095098, 385405617, -868708176, 1959010348, -4417777937, 9957188242, -22420045445
FORMULA
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A000219(k). G.f.: (1/(1 + x))*exp(Sum_{k>=1} sigma_2(k)*x^k/(k*(1 + x)^k)). - Ilya Gutkovskiy, Aug 20 2018
MATHEMATICA
nmax = 40; s = CoefficientList[Series[Product[1/(1-x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]; Table[Sum[(-1)^(n-k) * Binomial[n, k] * s[[k+1]], {k, 0, n}], {n, 0, nmax}]
Second Moebius transform of A000219. Number of plane partitions of n whose multiset of rows is aperiodic and whose multiset of columns is also aperiodic.
+20
4
1, 1, 1, 4, 8, 22, 34, 84, 137, 271, 450, 857, 1373, 2483, 3993, 6823, 10990, 18332, 28966, 47328, 74286, 118614, 184755, 290781, 448010, 695986, 1063773, 1632100, 2474970, 3759610, 5654233, 8512307, 12710995, 18973247, 28139285, 41690830, 61423271, 90379782
COMMENTS
A multiset is aperiodic if its multiplicities are relatively prime.
Also the number of plane partitions of n whose multiset of rows is aperiodic and whose parts are relatively prime.
FORMULA
The Moebius transform T of a sequence q is T(q)(n) = Sum_{d|n} mu(n/d) * q(d) where mu = A008683. The first Moebius transform of A000219 is A300275 and the third is A323585.
EXAMPLE
The a(4) = 8 plane partitions with aperiodic multisets of rows and columns:
4 31 211
.
3 21 111
1 1 1
.
2 11
1 1
1 1
The a(4) = 8 plane partitions with aperiodic multiset of rows and relatively prime parts:
31 211 1111
.
3 21 111
1 1 1
.
2 11
1 1
1 1
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS, Join@@Permutations/@facs[n], {2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y], And[GCD@@Length/@Split[#]==1, And@@GreaterEqual@@@#, And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]], {y, Select[IntegerPartitions[n], GCD@@#==1&]}], {n, 10}]
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