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A023871 Expansion of Product_{k>=1} (1 - x^k)^(-k^2). 26
1, 1, 5, 14, 40, 101, 266, 649, 1593, 3765, 8813, 20168, 45649, 101591, 223654, 486046, 1045541, 2225167, 4692421, 9804734, 20318249, 41766843, 85218989, 172628766, 347338117, 694330731, 1379437080, 2724353422, 5350185097, 10449901555, 20304465729, 39254599832 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

In general, if g.f. = Product_{k>=1} 1/(1 - x^k)^(c2*k^2 + c1*k + c0) and c2 > 0, then a(n) ~ exp(4*Pi * c2^(1/4) * n^(3/4) / (3*15^(1/4)) + c1*Zeta(3) / Pi^2 * sqrt(15*n/c2) + (Pi * 5^(1/4) * c0 / (2*3^(3/4) * c2^(1/4)) - 15^(5/4) * c1^2 * Zeta(3)^2 / (2*c2^(5/4) * Pi^5)) * n^(1/4) + c1/12 + 75 * c1^3 * Zeta(3)^3 / (c2^2 * Pi^8) - 5*c0 * c1 * Zeta(3) / (4*c2 * Pi^2) - c2*Zeta(3) / (4*Pi^2)) * Pi^(c1/12) * (c2/15)^(1/8 + c0/8 + c1/48) / (A^c1 * 2^((c0 + 3)/2) * n^(5/8 + c0/8 + c1/48)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 08 2017

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 1001 terms from Alois P. Heinz)

G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 21.

FORMULA

a(n) = 1/n*Sum_{k=1..n} a(n-k)*sigma_3(k), n > 0, a(0)=1, where sigma_3(n) = A001158(n) = sum of cubes of divisors of n. - Vladeta Jovovic, Jan 20 2002

G.f.: Prod_{n>=1} exp(sigma_3(n)*x^n/n), where sigma_3(n) is the sum of cubes of divisors of n (=A001158(n)). - N-E. Fahssi, Mar 28 2010

G.f. (conjectured): 1/prod(n>=1, E(x^n)^J2(n))) where E(x) = prod(n>=1,1-x^n) and J2(n) = A007434(n). - Joerg Arndt, Jan 25 2011

a(n) ~ exp(4 * Pi * n^(3/4) / (3^(5/4) * 5^(1/4)) - Zeta(3) / (4*Pi^2)) / (2^(3/2) * 15^(1/8) * n^(5/8)), where Zeta(3) = A002117 = 1.2020569031595942853997... . - Vaclav Kotesovec, Feb 27 2015

MAPLE

with(numtheory):

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

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

    end:

seq(a(n), n=0..35); # Alois P. Heinz, Nov 02 2012

MATHEMATICA

max = 31; Series[ Product[ 1/(1-x^k)^k^2, {k, 1, max}], {x, 0, max}] // CoefficientList[#, x]& (* Jean-François Alcover, Mar 05 2013 *)

PROG

(PARI) m=40; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^2)) \\ G. C. Greubel, Oct 29 2018

(MAGMA) m:=40; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^2: k in [1..m]]) )); // G. C. Greubel, Oct 29 2018

(SageMath) # uses[EulerTransform from A166861]

b = EulerTransform(lambda n: n^2)

print([b(n) for n in range(32)]) # Peter Luschny, Nov 11 2020

CROSSREFS

Euler transform of squares (A000290).

Cf. A000219, A023872-A023878, A294530.

Column k=2 of A144048. - Alois P. Heinz, Nov 02 2012

Sequence in context: A119996 A027089 A184437 * A274598 A327682 A171185

Adjacent sequences:  A023868 A023869 A023870 * A023872 A023873 A023874

KEYWORD

nonn,changed

AUTHOR

Olivier Gérard

EXTENSIONS

Definition corrected by Franklin T. Adams-Watters and R. J. Mathar, Dec 04 2006

STATUS

approved

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Last modified November 24 01:45 EST 2020. Contains 338603 sequences. (Running on oeis4.)