A261568 Expansion of Product_{k>=1} (1 + 4*x^k).

1, 4, 4, 20, 20, 36, 100, 116, 180, 260, 580, 660, 1044, 1380, 2020, 3444, 4340, 6020, 8260, 11220, 14740, 23140, 28196, 38900, 50420, 67780, 85956, 114900, 157140, 197860, 257060, 331060, 423540, 540100, 687620, 864084, 1145300, 1406500, 1789860, 2231860

Offset: 0

Author

Vaclav Kotesovec, Aug 24 2015

Keywords

• nonn

Links

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

Crossrefs

Cf. A000009, A032302, A032308, A261569.

Programs

Maple

b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, 4*b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 24 2015

Mathematica

nmax = 40; CoefficientList[Series[Product[1 + 4*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*4^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
(QPochhammer[-4, x]/5 + O[x]^58)[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

Formula

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(5*Pi)*n^(3/4)), where c = Pi^2/6 + 2*log(2)^2 + polylog(2, -1/4) = 2.36993979699836583198553742535032304875... . - Vaclav Kotesovec, Jan 04 2016

G.f.: Sum_{i>=0} 4^i*x^(i*(i+1)/2)/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018