A304877 -id:A304877 - 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

A304873 G.f.: Sum_{k>=0} p(k)^4 * x^k / Sum_{k>=0} p(k)*x^k, where p(n) is the partition function A000041(n).

Original entry on oeis.org

1, 0, 14, 64, 528, 1696, 11616, 33600, 169072, 525760, 2069922, 5928066, 22259874, 59321760, 193797792, 526647420, 1566376990, 4012181104, 11456306798, 28263784110, 75995086336, 184440427360, 468750673616, 1104027571108, 2730165482640, 6239956155696

Offset: 0

Vaclav Kotesovec, May 20 2018

nonn

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

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A054440 (m=2), A260664 (m=3).

Cf. A000041, A001255, A133042, A304877, A304878.

nmax = 25; CoefficientList[Series[Sum[PartitionsP[k]^4*x^k, {k, 0, nmax}] / Sum[PartitionsP[k]*x^k, {k, 0, nmax}], {x, 0, nmax}], x]

a(n) ~ 2^(3/4) * 3^(3/2) * 5^(13/4) * exp(Pi*sqrt(10*n)) / (2^22 * n^(15/4)).

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)).

A304988 G.f.: Sum_{k>=0} A000041(k)^2 * x^k / Sum_{k>=0} A000009(k) * x^k.

Original entry on oeis.org

1, 0, 3, 4, 16, 20, 67, 84, 231, 324, 735, 1026, 2265, 3086, 6199, 8880, 16564, 23390, 42378, 59496, 103588, 146376, 244278, 344186, 564013, 788168, 1255201, 1758400, 2738833, 3812242, 5846114, 8092092, 12200957, 16848156, 24991705, 34365176, 50392543

Offset: 0

Vaclav Kotesovec, May 23 2018

nonn

Cf. A000009, A000041, A001255, A054440, A304877, A305350.

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

a(n) ~ 7^(3/2) * exp(Pi*sqrt(7*n/3)) / (768*n^2).

A304987 G.f.: Sum_{k>=0} A000041(k) * x^k / Sum_{k>=0} A000009(k)^2 * x^k.

Original entry on oeis.org

1, 0, 1, -2, 2, -6, 3, -16, 17, -34, 47, -78, 153, -178, 373, -530, 954, -1410, 2280, -3896, 5908, -9988, 15170, -25908, 40659, -65136, 105967, -169056, 276483, -435624, 712052, -1139814, 1839535, -2955466, 4745201, -7689672, 12303439, -19866340, 31904000

Offset: 0

Vaclav Kotesovec, May 23 2018

sign

Cf. A000009, A000041, A054440, A304877, A304990.

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

a(n) ~ (-1)^n * c * d^n, where d = 1.6096199212376592810929072080593393678131347423108390218672748044914523428584..., c = 0.4455996319406557616008349987776746416976798533740571426884585957313974660...

A304989 G.f.: Sum_{k>=0} A000041(k)^2 * x^k / Sum_{k>=0} A000009(k)^2 * x^k.

Original entry on oeis.org

1, 0, 3, 2, 16, 10, 59, 32, 187, 90, 519, 152, 1439, 164, 3525, -246, 8904, -2500, 21748, -10836, 53918, -36508, 131424, -115266, 328703, -336608, 812615, -957464, 2046225, -2634166, 5152190, -7145682, 13121677, -19039178, 33473773, -50395004, 86035125

Offset: 0

Vaclav Kotesovec, May 23 2018

sign

Cf. A000009, A000041, A001255, A054440, A304877, A304990.

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

a(n) ~ (-1)^n * c * d^n, where d = 1.6096199212376592810929072080593393678131347423108390218672748044914523428584..., c = 3.049014588253509415528984781833089943634060493523166258285691300445092167...

cp's OEIS Frontend

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

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A304873 G.f.: Sum_{k>=0} p(k)^4 * x^k / Sum_{k>=0} p(k)*x^k, where p(n) is the partition function A000041(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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A304988 G.f.: Sum_{k>=0} A000041(k)^2 * x^k / Sum_{k>=0} A000009(k) * x^k.

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A304987 G.f.: Sum_{k>=0} A000041(k) * x^k / Sum_{k>=0} A000009(k)^2 * x^k.

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A304989 G.f.: Sum_{k>=0} A000041(k)^2 * x^k / Sum_{k>=0} A000009(k)^2 * x^k.

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