A261562 Expansion of Product_{k>=1} (1 + 2*x^k)^k.

1, 2, 4, 14, 24, 58, 124, 238, 480, 922, 1764, 3238, 6008, 10794, 19292, 34166, 59504, 103042, 176452, 299958, 505240, 845570, 1403324, 2315118, 3794640, 6180370, 10009540, 16121374, 25829512, 41171690, 65320956, 103140062, 162149488, 253823178, 395698276

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 2001 terms from Vaclav Kotesovec)

Formula

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{d|n} -(-2)^d * n^2/d^2 ). - Paul D. Hanna, Sep 30 2015

a(n) ~ c^(1/6) * exp(3^(2/3)*c^(1/3)*n^(2/3)/2) / (3^(3/4)*sqrt(2*Pi)*n^(2/3)), where c = Pi^2*log(2) + log(2)^3 - 6*polylog(3, -1/2) = 10.00970018379942727227807189532511265744588249928680712064... . - Vaclav Kotesovec, Jan 04 2016

Maple

b:= proc(n, i) option remember;  `if`(n=0, 1, `if`(i<1, 0,
      add(2^j*binomial(i, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 21 2018

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 + 2*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*2^k/k*x^k/(1 - x^k)^2, {k, 1, nmax}]], {x, 0, nmax}], x]
nmax = 50; s = 1+2*x; Do[s*=Sum[Binomial[k, j]*2^j*x^(j*k), {j, 0, nmax/k}]; s = Take[Expand[s], Min[nmax + 1, Exponent[s, x] + 1]]; , {k, 2, nmax}]; CoefficientList[s, x] (* Vaclav Kotesovec, Jan 08 2016 *)

Prog

(PARI) {a(n) = polcoeff( exp( sum(m=1, n, x^m/m * sumdiv(m, d, -(-2)^d * m^2/d^2) ) +x*O(x^n)), n)}
for(n=0, 40, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 30 2015

Crossrefs

Cf. A026007, A032302, A261561, A261563, A266857, A266891.

Sequence in context: A090889 A350050 A096012 * A102930 A357996 A135113

Adjacent sequences: A261559 A261560 A261561 * A261563 A261564 A261565

Keyword

nonn

Author

Vaclav Kotesovec, Aug 24 2015

Status

approved