A294655 -id:A294655 - cp's OEIS frontend

A294591 Expansion of Product_{k>=1} 1/((1 - x^(2k-1))^(k(3k-1)/2)(1 - x^(2k))^(k(3*k+1)/2)).

1, 1, 3, 8, 18, 40, 88, 184, 384, 783, 1573, 3110, 6087, 11745, 22450, 42466, 79597, 147890, 272632, 498696, 905846, 1634270, 2929804, 5220581, 9249440, 16297659, 28567571, 49825296, 86487331, 149438681, 257077485, 440378787, 751313413, 1276765557, 2161511352

Offset: 0

Ilya Gutkovskiy, Nov 05 2017

nonn

Euler transform of the generalized pentagonal numbers (A001318).

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Pentagonal Number

Cf. A000294, A001318, A095699, A278768, A294654, A294655, A294667, A294669.

nmax = 34; CoefficientList[Series[Product[1/((1 - x^(2 k - 1))^(k (3 k - 1)/2) (1 - x^(2 k))^(k (3 k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Ceiling[d/2] Ceiling[(3 d + 1)/2]/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 34}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A001318(k).

a(n) ~ exp(Pi * 2^(5/4) / (3*5^(1/4)) * n^(3/4) + 3*Zeta(3) * sqrt(5*n) / (2^(3/2) * Pi^2) + (Pi/48 - 45*Zeta(3)^2 / (8*Pi^5)) * (5*n/2)^(1/4) + 225*Zeta(3)^3 / (8*Pi^8) - 11*Zeta(3) / (64*Pi^2))/ (2^(95/48) * 5^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 07 2017

A294654 Expansion of Product_{k>=1} 1/((1 - x^(2k-1))^(k(5k-3)/2)(1 - x^(2k))^(k(5*k+3)/2)).

Original entry on oeis.org

1, 1, 5, 12, 35, 81, 208, 475, 1123, 2505, 5617, 12192, 26368, 55797, 117255, 242660, 498126, 1010515, 2033662, 4053214, 8017622, 15729219, 30643069, 59268267, 113898873, 217480476, 412813600, 779042099, 1462188257, 2729852845, 5070966794, 9373909586, 17247473718

Offset: 0

Ilya Gutkovskiy, Nov 06 2017

nonn

Euler transform of the generalized heptagonal numbers (A085787).

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Heptagonal Number

Cf. A000294, A085787, A278769, A294591, A294621, A294655.

nmax = 32; CoefficientList[Series[Product[1/((1 - x^(2 k - 1))^(k (5 k - 3)/2) (1 - x^(2 k))^(k (5 k + 3)/2)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (5 d (d + 1)/8 + (-1)^d (2 d + 1)/16 - 1/16), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A085787(k).

a(n) ~ exp(Pi * (2/3)^(5/4) * n^(3/4) + 5*Zeta(3) * sqrt(3*n) / (2^(3/2) * Pi^2) - (75*3^(1/4) * Zeta(3)^2 / (2^(13/4) * Pi^5) + Pi / (2^(17/4) * 3^(3/4))) * n^(1/4) + 375 * Zeta(3)^3 / (8*Pi^8) - 5*Zeta(3) / (64*Pi^2) + 1/12) * Pi^(1/12) / (A * 2^(11/6) * 3^(7/48) * n^(31/48)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 07 2017

A294691 Expansion of Product_{k>=1} 1 / (1 - x^(2k - 1))^(k(3*k - 2)).

Original entry on oeis.org

1, 1, 1, 9, 9, 30, 66, 106, 274, 459, 1010, 1862, 3552, 6973, 12446, 24245, 43041, 80372, 144482, 259633, 468047, 822642, 1468714, 2556542, 4493704, 7782441, 13470564, 23204471, 39679759, 67855411, 115004992, 194984378, 328183865, 551595570, 922663665

Offset: 0

Vaclav Kotesovec, Nov 07 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A294655, A294668, A294669.

nmax = 40; CoefficientList[Series[Product[1/(1-x^(2*k-1))^(k*(3*k-2)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi * 2^(5/4) / (3*5^(1/4)) * n^(3/4) + Zeta(3) * sqrt(5*n) / (Pi^2 * sqrt(2)) - (5*Zeta(3)^2 / (2*Pi^5) + Pi/24) * (5*n/2)^(1/4) + 25*Zeta(3)^3 / (3*Pi^8) + 2*Zeta(3) / (3*Pi^2) - 1/24) * sqrt(A) / (2^(173/96) * 5^(11/96) * Pi^(1/24) * n^(59/96)), where A is the Glaisher-Kinkelin constant A074962.

A294692 Expansion of Product_{k>=1} 1 / (1 - x^k)^(k(3k + 2)).

Original entry on oeis.org

1, 5, 31, 148, 667, 2754, 10823, 40393, 145085, 502780, 1690603, 5530649, 17658430, 55141520, 168751779, 506933980, 1496999360, 4350994324, 12460305177, 35192973824, 98116587875, 270220568883, 735668636567, 1981082952258, 5279879097853, 13933764841202

Offset: 0

Vaclav Kotesovec, Nov 07 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..5000

Cf. A074962, A274998, A278768, A294655, A294667.

N:= 50:
S:= series(mul(1/(1-x^k)^(k*(3*k+2)), k=1..N),x,N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Nov 07 2017

nmax = 30; CoefficientList[Series[Product[1/(1-x^k)^(k*(3*k+2)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(4*Pi*n^(3/4) / (3*5^(1/4)) + 2*Zeta(3) * sqrt(5*n) / Pi^2 - 2*5^(5/4) * Zeta(3)^2 * n^(1/4) / Pi^5 + 200*Zeta(3)^3 / (3*Pi^8) - 3*Zeta(3) / (4*Pi^2) + 1/6) * Pi^(1/6) / (A^2 * 2^(3/2) * 5^(1/6) * n^(2/3)), where A is the Glaisher-Kinkelin constant A074962.

cp's OEIS Frontend

A294591 Expansion of Product_{k>=1} 1/((1 - x^(2k-1))^(k(3k-1)/2)(1 - x^(2k))^(k(3*k+1)/2)).

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A294654 Expansion of Product_{k>=1} 1/((1 - x^(2k-1))^(k(5k-3)/2)(1 - x^(2k))^(k(5*k+3)/2)).

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A294691 Expansion of Product_{k>=1} 1 / (1 - x^(2k - 1))^(k(3*k - 2)).

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A294692 Expansion of Product_{k>=1} 1 / (1 - x^k)^(k(3k + 2)).

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cp's OEIS Frontend

A294591 Expansion of Product_{k>=1} 1/((1 - x^(2*k-1))^(k*(3*k-1)/2)*(1 - x^(2*k))^(k*(3*k+1)/2)).

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A294654 Expansion of Product_{k>=1} 1/((1 - x^(2*k-1))^(k*(5*k-3)/2)*(1 - x^(2*k))^(k*(5*k+3)/2)).

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A294691 Expansion of Product_{k>=1} 1 / (1 - x^(2*k - 1))^(k*(3*k - 2)).

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A294692 Expansion of Product_{k>=1} 1 / (1 - x^k)^(k*(3*k + 2)).

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A294591 Expansion of Product_{k>=1} 1/((1 - x^(2k-1))^(k(3k-1)/2)(1 - x^(2k))^(k(3*k+1)/2)).

A294654 Expansion of Product_{k>=1} 1/((1 - x^(2k-1))^(k(5k-3)/2)(1 - x^(2k))^(k(5*k+3)/2)).

A294691 Expansion of Product_{k>=1} 1 / (1 - x^(2k - 1))^(k(3*k - 2)).

A294692 Expansion of Product_{k>=1} 1 / (1 - x^k)^(k(3k + 2)).