A261561 Expansion of Product_{k>=1} (1/(1 - 2*x^k))^k.

1, 2, 8, 22, 64, 162, 424, 1022, 2480, 5770, 13336, 30046, 67184, 147554, 321592, 692278, 1479568, 3133474, 6596008, 13788606, 28679264, 59335530, 122256456, 250875550, 513116864, 1046190786, 2127557592, 4316282006, 8739096992, 17661731138, 35639764536

Offset: 0

Author

Vaclav Kotesovec, Aug 24 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 1001 terms from Vaclav Kotesovec)

Crossrefs

Cf. A000219, A070933, A246935, A261562, A261563, A261565.

Programs

Maple

b:= proc(n, i) option remember;  `if`(n=0, 1, `if`(i<1, 0,
      add(2^j*binomial(i+j-1, 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/(1 - 2*x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[Exp[Sum[2^k/k*x^k/(1 - x^k)^2, {k, 1, nmax}]], {x, 0, nmax}], x]

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

Formula

a(n) ~ c * 2^n, where c = Product_{j>=1} 1/(1 - 1/2^j)^(j+1) = 34.7387234654851595844514193757064296508992247003230539635669599773458896...

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