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A327050 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2*k)) * (1 + x^(3*k)) * (1 + x^(4*k)) * (1 + x^(5*k)) / ((1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(4*k)) * (1 - x^(5*k))). 3
1, 2, 6, 14, 32, 66, 136, 260, 494, 902, 1620, 2832, 4890, 8260, 13792, 22664, 36824, 59060, 93814, 147364, 229490, 354052, 541916, 822736, 1240292, 1856246, 2760368, 4078522, 5990900, 8749052, 12708920, 18363656, 26404386, 37783040, 53820120, 76324576 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Convolution of A327047 and A327044.

In general, for fixed m>=1, if g.f. = Product_{k>=1} (Product_{j=1..m} (1 + x^(j*k)) / (1 - x^(j*k))), then a(n) ~ sqrt(Gamma(m+1)) * HarmonicNumber(m)^((m+1)/4) * exp(Pi*sqrt(HarmonicNumber(m)*n)) / (2^(3*(m+1)/2) * n^((m+3)/4)).

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

FORMULA

a(n) ~ 137^(3/2) * exp(sqrt(137*n/15)*Pi/2) / (15*2^(21/2)*n^2).

MATHEMATICA

nmax = 50; CoefficientList[Series[Product[(1+x^k) * (1+x^(2*k)) * (1+x^(3*k)) * (1+x^(4*k)) * (1+x^(5*k)) / ((1-x^k) * (1-x^(2*k)) * (1-x^(3*k)) * (1-x^(4*k)) * (1-x^(5*k))), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A015128, A246584, A327048, A327049.

Cf. A301554.

Sequence in context: A055292 A327049 A035592 * A301554 A217941 A346679

Adjacent sequences: A327047 A327048 A327049 * A327051 A327052 A327053

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Aug 16 2019

STATUS

approved

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Last modified November 27 10:03 EST 2022. Contains 358367 sequences. (Running on oeis4.)