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A027998 Expansion of Product_{m>=1} (1+q^m)^(m^2). 34
1, 1, 4, 13, 31, 83, 201, 487, 1141, 2641, 5972, 13309, 29248, 63360, 135688, 287197, 601629, 1247909, 2565037, 5226816, 10565132, 21192569, 42202909, 83466925, 163999684, 320230999, 621579965, 1199659836, 2302765961, 4397132933, 8354234552, 15795913477 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

In general, if g.f. = Product_{k>=1} (1 + x^k)^(c2*k^2 + c1*k + c0) and c2 > 0, then a(n) ~ exp(2*Pi/3 * (14*c2/15)^(1/4) * n^(3/4) + 3*c1 * Zeta(3) / Pi^2 * sqrt(15*n/(14*c2)) + (Pi * c0 * (5/(14*c2))^(1/4) / (2*3^(3/4)) - 9*c1^2 * Zeta(3)^2 * (15/(14*c2))^(5/4) / Pi^5) * n^(1/4) + 2025 * c1^3 * Zeta(3)^3 / (49 * c2^2 * Pi^8) - 15*c0*c1*Zeta(3) / (28*c2 * Pi^2)) * ((7*c2)/15)^(1/8) / (2^(15/8 + c0/2 + c1/12) * n^(5/8)). - Vaclav Kotesovec, Nov 08 2017

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

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. 22.

FORMULA

a(n) = 1/n*Sum_{k=1..n} A078307(k)*a(n-k). - Vladeta Jovovic, Nov 22 2002

a(n) ~ 7^(1/8) * exp(2/3 * Pi * (14/15)^(1/4) * n^(3/4)) / (2^(15/8) * 15^(1/8) * n^(5/8)). - Vaclav Kotesovec, Mar 05 2015

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + x^k)/(k*(1 - x^k)^3)). - Ilya Gutkovskiy, May 30 2018

MAPLE

with(numtheory):

b:= proc(n) option remember;

      add((-1)^(n/d+1)*d^3, d=divisors(n))

    end:

a:= proc(n) option remember;

      `if`(n=0, 1, add(b(k)*a(n-k), k=1..n)/n)

    end:

seq(a(n), n=0..35);  # Alois P. Heinz, Aug 03 2013

MATHEMATICA

a[0] = 1; a[n_] := a[n] = 1/n*Sum[Sum[(-1)^(k/d+1)*d^3, {d, Divisors[k]}]*a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 31} ] (* Jean-Fran├žois Alcover, Jan 17 2014, after Vladeta Jovovic *)

nmax=50; CoefficientList[Series[Product[(1+x^k)^(k^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 05 2015 *)

PROG

(PARI) x = 'x + O('x ^ 50); Vec(prod(k=1, 50, (1 + x^k)^(k^2))) \\ Indranil Ghosh, Apr 05 2017

(MAGMA) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+x^k)^k^2: k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

CROSSREFS

Cf. A026007, A248882, A248883, A248884, A285224, A304049.

Column k=2 of A284992.

Sequence in context: A116411 A333047 A106337 * A026567 A218958 A226839

Adjacent sequences:  A027995 A027996 A027997 * A027999 A028000 A028001

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 20 01:04 EDT 2022. Contains 353847 sequences. (Running on oeis4.)