A278769 -id:A278769 - cp's OEIS frontend

A318118 a(n) = [x^n] exp(Sum_{k>=1} x^k(1 + (n - 3)x^k)/(k*(1 - x^k)^3)).

1, 1, 3, 10, 40, 150, 616, 2456, 10102, 41400, 171526, 712111, 2972115, 12434993, 52195414, 219567909, 925704792, 3909841659, 16541598215, 70085877919, 297347922785, 1263046810334, 5370930049915, 22861883482838, 97402827429118, 415332438952517, 1772380322197432

Offset: 0

Ilya Gutkovskiy, Aug 18 2018

nonn

For n > 2, a(n) is the n-th term of the Euler transform of n-gonal numbers.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms
Index to sequences related to polygonal numbers

Cf. A000294, A023871, A274998, A278767, A278768, A278769.

Table[SeriesCoefficient[Exp[Sum[x^k (1 + (n - 3) x^k)/(k (1 - x^k)^3), {k, 1, n}]], {x, 0, n}], {n, 0, 26}]

a(n) ~ c * d^n / sqrt(n), where d = 4.3505530790182509701639869563721679988879373943131559534408716195123... and c = 0.2276354216252041005336767937139336687746108521151301186102034... - Vaclav Kotesovec, Aug 18 2018

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

A317020 Expansion of Product_{k>=1} 1/(1 - x^k)^((5k-1)binomial(k+2,3)/4).

Original entry on oeis.org

1, 1, 10, 45, 185, 710, 2766, 10270, 37444, 132765, 462327, 1579563, 5311361, 17584084, 57414594, 185032557, 589183035, 1854974066, 5778722817, 17823440534, 54458411332, 164917654587, 495219323844, 1475145786950, 4360576440676, 12796007418881, 37287660835368, 107930276062786

Offset: 0

Ilya Gutkovskiy, Jul 19 2018

nonn

Euler transform of A002418.

N. J. A. Sloane, Transforms

Cf. A000391, A002418, A278769, A279218, A305653, A317017, A317019, A317021.

a:=series(mul(1/(1-x^k)^((5*k-1)*binomial(k+2,3)/4),k=1..100),x=0,28): seq(coeff(a,x,n),n=0..27); # Paolo P. Lava, Apr 02 2019

nmax = 27; CoefficientList[Series[Product[1/(1 - x^k)^((5 k - 1) Binomial[k + 2, 3]/4), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 27; CoefficientList[Series[Exp[Sum[x^k (1 + 4 x^k)/(k (1 - x^k)^5), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (d + 1) (d + 2) (5 d - 1)/24, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 27}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A002418(k).
G.f.: exp(Sum_{k>=1} x^k*(1 + 4*x^k)/(k*(1 - x^k)^5)).
a(n) ~ (5/7)^(703/8640)/(2 * 3^(2143/2880) * n^(5023/8640) * Pi^(17/1440)) * exp(-1/144 + (1/12-Zeta'(-1))/12 - (21 * Zeta(3))/(400 * Pi^2) + (62921 * Zeta(5))/(80000 * Pi^4) - (194481 * Zeta(3) * Zeta(5)^2)/(50 * Pi^12) - (200120949 * Zeta(5)^3)/(1250 * Pi^14) + (28594081676916 * Zeta(5)^5)/(3125 * Pi^24) + (7 * Zeta'(-3))/12 + ((-343 * (7/5)^(1/6) * Pi)/(96000 * sqrt(3)) + (147 * (7/5)^(1/6) * sqrt(3) * Zeta(3) * Zeta(5))/(10 * Pi^7) + (1058841 * (7/5)^(1/6) * sqrt(3) * Zeta(5)^2)/(2000 * Pi^9) - (18211006359 * (7/5)^(1/6) * sqrt(3) * Zeta(5)^4)/(500 * Pi^19)) * n^(1/6) + (-((7/5)^(1/3) * Zeta(3))/(4 * Pi^2) - (1029 * (7/5)^(1/3) * Zeta(5))/(200 * Pi^4) + (10890936 * (7/5)^(1/3) * Zeta(5)^3)/(25 * Pi^14)) * n^(1/3) + ((7 * sqrt(7/15) * Pi)/120 - (9261 * sqrt(21/5) * Zeta(5)^2)/(5 * Pi^9)) * sqrt(n) + ((63 * (7/5)^(2/3) * Zeta(5))/(2 * Pi^4)) * n^(2/3) + ((2 * sqrt(3) * Pi)/(5^(5/6) * 7^(1/6))) * n^(5/6)). - Vaclav Kotesovec, Jul 28 2018

cp's OEIS Frontend

A318118 a(n) = [x^n] exp(Sum_{k>=1} x^k(1 + (n - 3)x^k)/(k*(1 - x^k)^3)).

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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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A317020 Expansion of Product_{k>=1} 1/(1 - x^k)^((5k-1)binomial(k+2,3)/4).

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cp's OEIS Frontend

A318118 a(n) = [x^n] exp(Sum_{k>=1} x^k*(1 + (n - 3)*x^k)/(k*(1 - x^k)^3)).

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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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A317020 Expansion of Product_{k>=1} 1/(1 - x^k)^((5*k-1)*binomial(k+2,3)/4).

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Maple

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A318118 a(n) = [x^n] exp(Sum_{k>=1} x^k(1 + (n - 3)x^k)/(k*(1 - x^k)^3)).

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

A317020 Expansion of Product_{k>=1} 1/(1 - x^k)^((5k-1)binomial(k+2,3)/4).