A023871 -id:A023871 - cp's OEIS frontend

A304446 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^(n^2).

1, 1, 14, 255, 6460, 209405, 8287038, 387605491, 20930373880, 1281932464680, 87828985857380, 6656774777650459, 553068813860022264, 49988877225605011590, 4883606791114233989450, 512829418039842285746460, 57607740718731604241384432, 6893420862444517638234527039

Offset: 0

Vaclav Kotesovec, May 12 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..330

Cf. A008485, A023871, A304445.

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

a(n) ~ exp(n + 3/2) * n^(n - 1/2) / sqrt(2*Pi).

A278767 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(2k-1)).

Original entry on oeis.org

1, 1, 7, 22, 71, 206, 616, 1712, 4743, 12677, 33407, 86085, 218677, 546060, 1345840, 3271893, 7861239, 18670881, 43883904, 102112483, 235401947, 537869136, 1218743007, 2739566083, 6111766043, 13536683750, 29775945929, 65065819486, 141285315728, 304935221675, 654318376244, 1396166024244, 2963068779402

Offset: 0

Ilya Gutkovskiy, Nov 28 2016

nonn

Euler transform of the hexagonal numbers (A000384).

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Hexagonal Number
Index to sequences related to polygonal numbers

Cf. A000294, A000384, A000335, A023871.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d^2*(2*d-1), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 02 2016

nmax=32; CoefficientList[Series[Product[1/(1 - x^k)^(k (2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(2*k-1)).

a(n) ~ exp(-Zeta'(-1) - Zeta(3)/(2*Pi^2) - 75*Zeta(3)^3/(4*Pi^8) - 15^(5/4)*Zeta(3)^2/(2^(9/4)*Pi^5) * n^(1/4) - sqrt(15/2)*Zeta(3)/Pi^2 * sqrt(n) + 2^(9/4)*Pi/(3^(5/4)*5^(1/4)) * n^(3/4)) / (2^(67/48) * 15^(5/48) * Pi^(1/12) * n^(29/48)). - Vaclav Kotesovec, Dec 02 2016

A278769 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(5k-3)/2).

Original entry on oeis.org

1, 1, 8, 26, 88, 269, 843, 2456, 7115, 19892, 54756, 147355, 390517, 1017091, 2612670, 6617641, 16556913, 40933339, 100104289, 242276236, 580718077, 1379161494, 3247074738, 7581837910, 17564867853, 40388447308, 92206496318, 209069338580, 470944571003, 1054178579266, 2345477963043, 5188246121144, 11412352653001

Offset: 0

Ilya Gutkovskiy, Nov 28 2016

nonn

Euler transform of the heptagonal numbers (A000566).

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Heptagonal Number
Index to sequences related to polygonal numbers

Cf. A000294, A000566, A000335, A023871.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d^2*(5*d-3)/2, d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 02 2016

nmax=32; CoefficientList[Series[Product[1/(1 - x^k)^(k (5 k - 3)/2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(5*k-3)/2).

a(n) ~ exp(-3*Zeta'(-1)/2 - 5*Zeta(3)/(8*Pi^2) - 81*Zeta(3)^3/(2*Pi^8) - 3^(13/4)*Zeta(3)^2/(2^(7/4)*Pi^5) * n^(1/4) - 3^(3/2)*Zeta(3)/(sqrt(2)*Pi^2) * sqrt(n) + 2^(7/4)*Pi/3^(5/4) * n^(3/4)) / (2^(51/32) * 3^(3/32) * Pi^(1/8) * n^(19/32)). - Vaclav Kotesovec, Dec 02 2016

A282327 Expansion of exp( Sum_{n>=1} sigma_3(2n)x^n/n ) in powers of x.

Original entry on oeis.org

1, 9, 77, 534, 3320, 18933, 100770, 506697, 2428161, 11161765, 49469005, 212246744, 884491121, 3589900607, 14223638534, 55122970206, 209307080221, 779837798559, 2854660220661, 10278494869342, 36439277959593, 127311828611819, 438712861233581

Offset: 0

Seiichi Manyama, Mar 03 2017

nonn

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

Cf. exp( Sum_{n>=1} sigma_k(2*n)*x^n/n ): A182818 (k=1), A283224 (k=2), this sequence (k=3).

Cf. exp( Sum_{n>=1} sigma_3(m*n)*x^n/n ): A023871 (m=1), this sequence (m=2), A283244 (m=3).

a(n) = (1/n)*Sum_{k=1..n} sigma_3(2*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A283244 Expansion of exp( Sum_{n>=1} sigma_3(3n)x^n/n ) in powers of x.

Original entry on oeis.org

1, 28, 518, 7439, 90517, 972398, 9472190, 85145743, 715281840, 5668682493, 42691867112, 307312234334, 2124355701646, 14157081285263, 91250293831492, 570441761053192, 3466874635995098, 20526329624103412, 118608374492197651, 669949478060261642

Offset: 0

Seiichi Manyama, Mar 03 2017

nonn

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

Cf. exp( Sum_{n>=1} sigma_k(3*n)*x^n/n ): A182819 (k=1), A283238 (k=2), this sequence (k=3).

Cf. exp( Sum_{n>=1} sigma_3(m*n)*x^n/n ): A023871 (m=1), A282327 (m=2), this sequence (m=3).

a(n) = (1/n)*Sum_{k=1..n} sigma_3(3*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A294778 Expansion of Product_{k>=1} 1/(1 - x^(2k-1))^(k(k-1)/2).

Original entry on oeis.org

1, 0, 0, 1, 0, 3, 1, 6, 3, 11, 12, 18, 29, 33, 69, 67, 138, 141, 275, 306, 516, 656, 972, 1353, 1828, 2712, 3477, 5280, 6654, 10038, 12756, 18789, 24369, 34796, 46167, 63990, 86629, 117189, 160698, 213984, 295092, 389517, 536683, 706590, 968289, 1276310

Offset: 0

Vaclav Kotesovec, Nov 08 2017

nonn

Cf. A023871, A035528, A258349, A294750.

nmax = 50; CoefficientList[Series[Product[1/(1-x^(2*k-1))^(k*(k-1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(2*Pi * n^(3/4) / (3^(5/4) * 5^(1/4)) - 5^(1/4) * Pi * n^(1/4) / (16*3^(3/4)) + 3*Zeta(3) / (32*Pi^2)) / (2^(31/16) * 15^(1/8) * n^(5/8)).

A302449 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(4k^2-1)/3).

Original entry on oeis.org

1, 1, 11, 46, 185, 700, 2676, 9646, 34166, 117500, 396506, 1310527, 4258313, 13607309, 42846151, 133039791, 407833188, 1235202869, 3699140386, 10960888382, 32154531807, 93437164720, 269087234273, 768340525743, 2176098269286, 6115444177489, 17058887661133

Offset: 0

Ilya Gutkovskiy, Apr 08 2018

nonn

Euler transform of A000447.

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms

Cf. A000335, A000447, A023871, A279215.

nmax = 26; CoefficientList[Series[Product[1/(1 - x^k)^(k (4 k^2 - 1)/3), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (4 d^2 - 1)/3, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 26}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A000447(k).

a(n) ~ exp(5 * Zeta(5)^(1/5) * n^(4/5)/2 - Zeta(3) * n^(2/5) / (12 * Zeta(5)^(2/5)) + 4*Zeta'(-3)/3 - 1/36 - Zeta(3)^2 / (720*Zeta(5))) * A^(1/3) * Zeta(5)^(83/900) / (2^(7/180) * sqrt(5*Pi) * n^(533/900)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 08 2018

A305654 a(n) = [x^n] exp(Sum_{k>=1} x^k(1 + x^k)/(k(1 - x^k)^n)).

Original entry on oeis.org

1, 1, 4, 14, 65, 323, 1890, 12002, 83901, 630818, 5081318, 43546333, 395422430, 3788368227, 38151667046, 402516707510, 4436230390977, 50948789415297, 608433141666219, 7540823673023319, 96826154085714992, 1285991546051286085, 17640769457638701839, 249602608552024560609

Offset: 0

Ilya Gutkovskiy, Jun 07 2018

nonn

N. J. A. Sloane, Transforms

Cf. A000990, A023871, A253289, A279215, A293554, A305206, A305653, A305655.

Table[SeriesCoefficient[Exp[Sum[x^k (1 + x^k)/(k (1 - x^k)^n), {k, 1, n}]], {x, 0, n}], {n, 0, 23}]
Table[SeriesCoefficient[Product[1/(1 - x^k)^(2 Binomial[n + k - 2, n - 1] - Binomial[n + k - 3, n - 2]), {k, 1, n}], {x, 0, n}], {n, 0, 23}]

a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^(2*binomial(n+k-2,n-1)-binomial(n+k-3,n-2)).

cp's OEIS Frontend

A304446 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^(n^2).

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A278767 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(2*k-1)).

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A278769 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(5*k-3)/2).

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A282327 Expansion of exp( Sum_{n>=1} sigma_3(2*n)*x^n/n ) in powers of x.

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A283244 Expansion of exp( Sum_{n>=1} sigma_3(3*n)*x^n/n ) in powers of x.

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A294778 Expansion of Product_{k>=1} 1/(1 - x^(2*k-1))^(k*(k-1)/2).

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A302449 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(4*k^2-1)/3).

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A305654 a(n) = [x^n] exp(Sum_{k>=1} x^k*(1 + x^k)/(k*(1 - x^k)^n)).

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Formula

A278767 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(2k-1)).

A278769 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(5k-3)/2).

A282327 Expansion of exp( Sum_{n>=1} sigma_3(2n)x^n/n ) in powers of x.

A283244 Expansion of exp( Sum_{n>=1} sigma_3(3n)x^n/n ) in powers of x.

A294778 Expansion of Product_{k>=1} 1/(1 - x^(2k-1))^(k(k-1)/2).

A302449 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(4k^2-1)/3).

A305654 a(n) = [x^n] exp(Sum_{k>=1} x^k(1 + x^k)/(k(1 - x^k)^n)).