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A317019 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*binomial(k+2,3)). 3
1, 1, 9, 39, 155, 570, 2131, 7599, 26667, 90996, 305144, 1004173, 3254123, 10385884, 32704819, 101678860, 312435675, 949498206, 2855953018, 8507079361, 25108844890, 73468004480, 213201630328, 613871526178, 1754365814430, 4978113020152, 14029639217532, 39281646364737 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Euler transform of A002417.

LINKS

Table of n, a(n) for n=0..27.

N. J. A. Sloane, Transforms

FORMULA

G.f.: Product_{k>=1} 1/(1 - x^k)^A002417(k).

G.f.: exp(Sum_{k>=1} x^k*(1 + 3*x^k)/(k*(1 - x^k)^5)).

a(n) ~ 1/(2^(601/720) * 3^(359/480) * 7^(119/1440) * n^(839/1440) * Pi^(1/240)) * exp(-Zeta(3)/(12 * Pi^2) + (491 * Zeta(5))/(400 * Pi^4) - (2250423 * Zeta(5)^3)/(10 * Pi^14) + (103355177121 * Zeta(5)^5)/(10 * Pi^24) + Zeta'(-3)/2 + ((-7 * 7^(1/6) * Pi)/(1200 * 2^(1/3) * sqrt(3)) + (27783 * sqrt(3) * 7^(1/6) * Zeta(5)^2)/(40 * 2^(1/3) * Pi^9) - (614365479 * sqrt(3) * 7^(1/6) * Zeta(5)^4)/(16 * 2^(1/3) * Pi^19)) * n^(1/6) + ((-63 * 7^(1/3) * Zeta(5))/(10 * 2^(2/3) * Pi^4) + (214326 * 14^(1/3) * Zeta(5)^3)/Pi^14) * n^(1/3) + ((sqrt(7/3) * Pi)/30 - (1701 * sqrt(21) * Zeta(5)^2)/(2 * Pi^9)) * sqrt(n) + ((27 * 7^(2/3) * Zeta(5))/(2 * 2^(1/3) * Pi^4)) * n^(2/3) + ((2 * 2^(1/3) * sqrt(3) * Pi)/(5 * 7^(1/6))) * n^(5/6)). - Vaclav Kotesovec, Jul 28 2018

MAPLE

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

MATHEMATICA

nmax = 27; CoefficientList[Series[Product[1/(1 - x^k)^(k Binomial[k + 2, 3]), {k, 1, nmax}], {x, 0, nmax}], x]

nmax = 27; CoefficientList[Series[Exp[Sum[x^k (1 + 3 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^3 (d + 1) (d + 2)/6, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 27}]

CROSSREFS

Cf. A000391, A002417, A278767, A279217, A305653, A317017, A317020, A317021.

Sequence in context: A212143 A294845 A124851 * A124041 A264085 A126396

Adjacent sequences:  A317016 A317017 A317018 * A317020 A317021 A317022

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Jul 19 2018

STATUS

approved

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Last modified January 25 03:49 EST 2020. Contains 331241 sequences. (Running on oeis4.)