A304873 -id:A304873 - cp's OEIS frontend

A054440 Number of ordered pairs of partitions of n with no common parts.

1, 0, 2, 4, 12, 16, 48, 60, 148, 220, 438, 618, 1302, 1740, 3216, 4788, 8170, 11512, 19862, 27570, 45448, 64600, 100808, 141724, 223080, 307512, 465736, 652518, 968180, 1334030, 1972164, 2691132, 3902432, 5347176, 7611484, 10358426, 14697028, 19790508, 27691500

Offset: 0

Herbert S. Wilf, May 13 2000

			a(3)=4 because of the 4 pairs of partitions of 3: (3,21),(3,111),(21,3),(111,3).

Reinhard Zumkeller, Table of n, a(n) for n = 0..5000
Sylvie Corteel, Carla D. Savage, Herbert S. Wilf, Doron Zeilberger, A pentagonal number sieve, J. Combin. Theory Ser. A 82 (1998), no. 2, 186-192.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
Wikipedia, Pentagonal number theorem

Cf. A000041, A001255, A001318, A087960, A260672, A260664, A260669, A304873, A304877.

Main diagonal of A284592.

a054440 = sum . zipWith (*) a087960_list . map a001255 . a260672_row
-- Reinhard Zumkeller, Nov 15 2015

with(combinat): p1 := sum(numbpart(n)^2*x^n, n=0..500): it := p1*product((1-x^i), i=1..500): s := series(it, x, 500): for i from 0 to 100 do printf(`%d,`,coeff(s,x,i)) od:

nmax = 50; CoefficientList[Series[Sum[PartitionsP[k]^2*x^k, {k, 0, nmax}]/Sum[PartitionsP[k]*x^k, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 04 2016 *)

G.f.: Sum[p(n)^2*x^n]/Sum[p(n)*x^n], with p(n)=number of partitions of n.
a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n)) / (64 * 2^(1/4) * n^(7/4)). - Vaclav Kotesovec, May 20 2018
a(n) = [(x*y)^n] Product_{k>=1} (1 + x^k / (1 - x^k) + y^k / (1 - y^k)). - Ilya Gutkovskiy, Apr 24 2025

Corrected and extended by James Sellers, May 23 2000

A260664 Number of ordered triples of partitions of n with no common parts.

Original entry on oeis.org

1, 0, 6, 18, 90, 192, 864, 1710, 5970, 13110, 36810, 75984, 210546, 410130, 1003908, 2045808, 4616730, 8950176, 19746720, 37297710, 78247344, 147410640, 294299424, 543058032, 1067679540, 1925323308, 3653769792, 6555529158, 12129597486, 21348640230

Offset: 0

Reinhard Zumkeller, Nov 15 2015

nonn

			a(3) = 18 because of the 18 triples of partitions of 3: (3,3,21), (3,3,111), (3,21,3), (3,21,21), (3,21,111), (3,111,3), (3,111,21), (3,111,111), (21,3,3), (21,3,21), (21,3,111), (21,21,3), (21,111,3), (111,3,3), (111,3,21), (111,3,111), (111,21,3) and (111,111,3);
a(3) = A000041(3-A001318(0))^3 - A000041(3-A001318(1))^3 - A000041(3-A001318(2))^3 = 3^3 - 2^3 - 1^3 = 27 - 8 - 1 = 18.

Reinhard Zumkeller, Table of n, a(n) for n = 0..5000
Sylvie Corteel, Carla D. Savage, Herbert S. Wilf, Doron Zeilberger, A pentagonal number sieve, J. Combin. Theory Ser. A 82 (1998), no. 2, 186-192.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
Wikipedia, Pentagonal number theorem

Cf. A000041, A133042, A001318, A087960, A260672, A054440, A304873, A304878.

a260664 = sum . zipWith (*) a087960_list . map a133042 . a260672_row

Table[Sum[(Cos[Pi*j/2] - Sin[Pi*j/2]) * PartitionsP[n - ((6*j^2 + 6*j + 1)/16 - (2*j + 1)*(-1)^j/16)]^3, {j, 0, Floor[Sqrt[8*n/3]]}], {n, 0, 30}] (* Vaclav Kotesovec, Jul 04 2016 *)
nmax = 50; CoefficientList[Series[Sum[PartitionsP[k]^3*x^k, {k, 0, nmax}] / Sum[PartitionsP[k]*x^k, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 04 2016 *)

a(n) = p(n)^3 - p(n-k(1))^3 - p(n-k(2))^3 + p(n-k(3))^3 + p(n-k(4))^3 - p(n-k(5))^3 - ..., with p=A000041 and k=A001318, see Wilf link: p. 2, (3).
G.f.: Sum[p(n)^3*x^n]/Sum[p(n)*x^n], with p(n)=number of partitions of n. - Vaclav Kotesovec, Jul 04 2016
a(n) ~ 2^(3/2) * exp(4*Pi*sqrt(n/3)) / (729 * 3^(1/4) * n^(11/4)). - Vaclav Kotesovec, May 20 2018

A304877 G.f.: Sum_{k>=0} q(k)^2 * x^k / Sum_{k>=0} q(k)*x^k, where q(n) is A000009(n).

Original entry on oeis.org

1, 0, 0, 2, 0, 4, 4, 8, 4, 20, 20, 28, 38, 52, 80, 128, 128, 176, 300, 316, 476, 648, 832, 972, 1428, 1720, 2340, 3014, 3844, 4588, 6556, 7476, 9760, 12588, 15596, 19480, 25140, 29796, 37728, 47604, 58140, 70856, 90148, 107692, 133228, 167284, 198692, 242728

Offset: 0

Vaclav Kotesovec, May 20 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000009, A054440, A304873, A304878.

nmax = 50; CoefficientList[Series[Sum[PartitionsQ[k]^2*x^k, {k, 0, nmax}] / Sum[PartitionsQ[k]*x^k, {k, 0, nmax}], {x, 0, nmax}], x]

a(n) ~ sqrt(3) * exp(Pi*sqrt(n)) / (2^(11/2) * n^(3/2)).

A304878 G.f.: Sum_{k>=0} q(k)^3 * x^k / Sum_{k>=0} q(k)*x^k, where q(n) is A000009(n).

Original entry on oeis.org

1, 0, 0, 6, 0, 18, 30, 60, 66, 258, 402, 606, 1266, 1866, 3744, 6864, 9648, 15432, 30510, 41166, 72342, 118140, 178800, 266262, 441462, 652164, 1006410, 1567692, 2309958, 3385554, 5321838, 7530462, 11128440, 16799958, 23916474, 35123964, 51357318, 72495126

Offset: 0

Vaclav Kotesovec, May 20 2018

nonn

In general, if m > 1 and g.f. = Sum_{k>=0} q(k)^m * x^k / Sum_{k>=0} q(k)*x^k, then a(n, m) ~ exp(Pi*sqrt((m^2 - 1)*n/3)) * (m^2 - 1)^(3*m/4 - 1/2) / (2^(2*m - 1/2) * 3^(m/4) * m^(3*m/2 - 1) * n^(3*m/4)).

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000009, A260664, A304873, A304877.

nmax = 50; CoefficientList[Series[Sum[PartitionsQ[k]^3*x^k, {k, 0, nmax}] / Sum[PartitionsQ[k]*x^k, {k, 0, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (81*6^(1/4)*n^(9/4)).

A304992 G.f.: Sum_{k>=0} A000041(k)^3 * x^k / Sum_{k>=0} A000009(k) * x^k.

Original entry on oeis.org

1, 0, 7, 18, 98, 210, 969, 1938, 7037, 15258, 44815, 93180, 262391, 518550, 1311015, 2657328, 6189160, 12124098, 27239760, 52063668, 111630480, 211503288, 432900236, 806091180, 1610854427, 2940167268, 5691072911, 10289144976, 19402974147, 34523231688

Offset: 0

Vaclav Kotesovec, May 23 2018

nonn

In general, if m > 1 and g.f. = Sum_{k>=0} A000041(k)^m * x^k / Sum_{k>=0} A000009(k) * x^k, then a(n, m) ~ exp(Pi*sqrt((2*m^2 - 1)*n/3)) * ((2*m^2 - 1)^(m - 1/2) / (2^(3*m - 1) * 3^(m/2) * m^(2*m - 1) * n^m)).

Cf. A000009, A000041, A133042, A260664, A304873, A304878, A304988.

nmax = 50; CoefficientList[Series[Sum[PartitionsP[k]^3*x^k, {k, 0, nmax}] / Sum[PartitionsQ[k]*x^k, {k, 0, nmax}], {x, 0, nmax}], x]

a(n) ~ 289 * sqrt(17/3) * exp(Pi*sqrt(17*n/3)) / (186624*n^3).

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A054440 Number of ordered pairs of partitions of n with no common parts.

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A260664 Number of ordered triples of partitions of n with no common parts.

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A304877 G.f.: Sum_{k>=0} q(k)^2 * x^k / Sum_{k>=0} q(k)*x^k, where q(n) is A000009(n).

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A304878 G.f.: Sum_{k>=0} q(k)^3 * x^k / Sum_{k>=0} q(k)*x^k, where q(n) is A000009(n).

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A304992 G.f.: Sum_{k>=0} A000041(k)^3 * x^k / Sum_{k>=0} A000009(k) * x^k.

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