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A274998 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(3*k-2)). 5
1, 1, 9, 30, 106, 339, 1106, 3355, 10102, 29358, 83908, 234394, 644286, 1739933, 4631675, 12153197, 31485413, 80576160, 203902261, 510490213, 1265353568, 3106771717, 7559844833, 18239351931, 43650061720, 103657177941, 244346681972, 571930478187, 1329655624297, 3071230379625, 7049750442386, 16085170634548, 36489192684910 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Euler transform of the octagonal numbers (A000567).

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

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, Octagonal Number

Index to sequences related to polygonal numbers

FORMULA

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

a(n) ~ exp(4*Pi*n^(3/4) / (3*5^(1/4)) - 2*Zeta(3) * sqrt(5*n) / Pi^2 - 10*Zeta(3)^2 * (5*n)^(1/4) / Pi^5 - 200*Zeta(3)^3 / (3*Pi^8) - 3*Zeta(3) / (4*Pi^2) - 1/6) * A^2 / (2^(3/2) * 5^(1/12) * Pi^(1/6) * n^(7/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 08 2017

MAPLE

with(numtheory):

a:= proc(n) option remember; `if`(n=0, 1, add(add(

      d^2*(3*d-2), d=divisors(j))*a(n-j), j=1..n)/n)

    end:

seq(a(n), n=0..35);  # Alois P. Heinz, Dec 02 2016

MATHEMATICA

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

PROG

(Python)

from sympy import divisors

from sympy.core.cache import cacheit

@cacheit

def a(n): return 1 if n==0 else sum([sum([d**2*(3*d - 2) for d in divisors(j)])*a(n - j) for j in xrange(1, n + 1)])/n

print map(a, xrange(51)) # Indranil Ghosh, Aug 06 2017, after Maple code

CROSSREFS

Cf. A000294, A000567, A000335, A023871, A278768, A294667, A294691, A294692.

Sequence in context: A301988 A212517 A319839 * A000440 A300643 A161684

Adjacent sequences:  A274995 A274996 A274997 * A274999 A275000 A275001

KEYWORD

nonn,easy

AUTHOR

Ilya Gutkovskiy, Nov 30 2016

STATUS

approved

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Last modified October 18 14:52 EDT 2018. Contains 316322 sequences. (Running on oeis4.)