A382977 Expansion of Product_{k>=1} 1/(1 - (2^k - 1) * x^k).

1, 1, 4, 11, 35, 87, 271, 659, 1908, 4832, 13132, 32688, 89109, 218385, 571489, 1427388, 3652877, 8980805, 22858201, 55822728, 140065621, 342001192, 845707856, 2052802367, 5057431745, 12197383588, 29738238996, 71604414162, 173406091548, 415167136507, 1000881376700

Offset: 0

Author

Seiichi Manyama, Apr 11 2025

Keywords

• nonn

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1, g(n) = 2^n - 1.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..600
• Seiichi Manyama, Generalized Euler transform.

Crossrefs

Cf. A322199, A382976, A382978.

Cf. A000225, A266964.

Programs

Magma

n := 30; R := PowerSeriesRing(Rationals(), n+1); f := &*[ 1 / (1 - (2^k - 1)*x^k) : k in [1..n] ]; coeffs := [Coefficient(f, i) : i in [0..n]]; coeffs; // Vincenzo Librandi, Apr 11 2025

Mathematica

n=30; CoefficientList[Normal@Series[Product[1/(1-(2^k-1) x^k),{k,1,n}],{x,0,n}],x] (* Vincenzo Librandi, Apr 11 2025 *)

PARI

f(n) = 1;
g(n) = 2^n-1;
a_vector(n) = my(b=vector(n, k, sumdiv(k, d, d*f(d)*g(d)^(k/d))), v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, b[j]*v[i-j+1])/i); v;

Formula

a(n) = Sum_{k=0..n} 2^k * A382974(k,n-k).

log(a(n)) ~ n*log(2) + Pi*sqrt(2*n/3). - Vaclav Kotesovec, Apr 13 2025