A292304 -id:A292304 - cp's OEIS frontend

A291698 a(n) = [x^n] Product_{k>=1} (1 + n*x^k).

1, 1, 2, 12, 20, 55, 294, 497, 1224, 2520, 14410, 21912, 54300, 104286, 220710, 1105215, 1697552, 3839382, 7356762, 14873580, 26275620, 132112596, 188666126, 423247104, 772560600, 1535398150, 2632049290, 4975242048, 21273166572, 30649985160, 64824339630, 116604788800, 223181224992

Offset: 0

Ilya Gutkovskiy, Aug 30 2017

The number of partitions of n into distinct parts where each part can be colored in n different ways. For example, there are 4 partitions of 6 into distinct parts, namely 6, 5 + 1, 4 + 2 and 3 + 2 + 1; allowing for the colorings gives a(6) = 6 + 6*6 + 6*6 + 6*6*6 = 294. - Peter Bala, Aug 31 2017

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..2000 from Robert Israel)

Main diagonal of A286957.

Cf. A022629, A124577, A292190, A292304.

seq(coeff(mul(1+n*x^k,k=1..n),x,n),n=0..50); # Robert Israel, Aug 30 2017

Table[SeriesCoefficient[Product[1 + n x^k, {k, 1, n}], {x, 0, n}], {n, 0, 32}]
Table[SeriesCoefficient[QPochhammer[-n, x]/(1 + n), {x, 0, n}], {n, 0, 32}]

a(n) = A286957(n,n).
a(n) == 0 (mod n); a(n) == n (mod n^2). - Peter Bala, Aug 31 2017
Conjecture: a(n) ~ exp(sqrt(2*(log(n)^2 + Pi^2/3)*n)) * (log(n)^2 + Pi^2/3)^(1/4) / (sqrt(Pi) * (2*n)^(5/4)). - Vaclav Kotesovec, Sep 15 2017

A292414 a(n) = [x^n] Product_{k>=1} (1 + 2^n*x^k).

Original entry on oeis.org

1, 2, 4, 72, 272, 2080, 270400, 2146432, 33751296, 403702272, 1103810790400, 17635156690944, 563431073648640, 13515197331283968, 360331952265379840, 37785849814204784082944, 1209091844251972299456512, 77374499118322174520328192, 3713890953695657990811811840

Offset: 0

Vaclav Kotesovec, Sep 16 2017

Vaclav Kotesovec, Table of n, a(n) for n = 0..183

Cf. A292304, A292415, A292416.

nmax = 20; Table[SeriesCoefficient[Product[(1+2^n*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]

{a(n)= polcoef(prod(k=1, n, (1+2^n*x^k +x*O(x^n))), n)};
for(n=0,20, print1(a(n), ", ")) \\ G. C. Greubel, Feb 02 2019

Conjecture: log(a(n)) ~ sqrt(2)*log(2)*n^(3/2). - Vaclav Kotesovec, Aug 22 2018

A292417 a(n) = [x^n] Product_{k>=1} 1/(1 - n^2*x^k).

Original entry on oeis.org

1, 1, 20, 819, 70160, 10188775, 2240751636, 692647082799, 286013768613952, 151994274055319070, 101020305070908050100, 82086758986568812837856, 80056656965795630400382608, 92282612223268812357487227077, 124113156850218393012451734737460

Offset: 0

Vaclav Kotesovec, Sep 16 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..214

Cf. A077335, A124577, A292304.

nmax = 20; Table[SeriesCoefficient[Product[1/(1-n^2*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]

{a(n)= polcoef(prod(k=1, n, 1/(1-n^2*x^k +x*O(x^n))), n)};
for(n=0,20, print1(a(n), ", ")) \\ G. C. Greubel, Feb 02 2019

a(n) ~ n^(2*n) * (1 + 1/n^2 + 2/n^4 + 3/n^6 + 5/n^8 + 7/n^10), for coefficients see A000041.

A292418 a(n) = [x^n] Product_{k>=1} (1 + n^2x^k) / (1 - n^2x^k).

Original entry on oeis.org

1, 2, 40, 1800, 149024, 21223800, 4609532520, 1414165715200, 581109518753920, 307788983933760954, 204081628466048180200, 165541724073121026987224, 161233041454793035411134240, 185663865439487951708529417080, 249499302292252719726304186789160

Offset: 0

Vaclav Kotesovec, Sep 16 2017

nonn

Convolution of A292304 and A292417.

G. C. Greubel, Table of n, a(n) for n = 0..214

Cf. A265758, A265844, A292304, A292417, A292419.

nmax = 20; Table[SeriesCoefficient[Product[(1+n^2*x^k)/(1-n^2*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]

{a(n)= polcoef(prod(k=1, n, ((1+n^2*x^k)/(1-n^2*x^k) +x*O(x^n))), n)};
for(n=0,20, print1(a(n), ", ")) \\ G. C. Greubel, Feb 02 2019

a(n) ~ 2 * n^(2*n) * (1 + 2/n^2 + 4/n^4 + 8/n^6 + 14/n^8 + 24/n^10), for coefficients see A015128.

cp's OEIS Frontend

A291698 a(n) = [x^n] Product_{k>=1} (1 + n*x^k).

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A292414 a(n) = [x^n] Product_{k>=1} (1 + 2^n*x^k).

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A292417 a(n) = [x^n] Product_{k>=1} 1/(1 - n^2*x^k).

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A292418 a(n) = [x^n] Product_{k>=1} (1 + n^2*x^k) / (1 - n^2*x^k).

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Formula

A292418 a(n) = [x^n] Product_{k>=1} (1 + n^2x^k) / (1 - n^2x^k).