A382978 Expansion of Product_{k>=1} (1 + (2^k - 1) * x^k).

1, 1, 3, 10, 22, 67, 160, 433, 986, 2774, 6386, 16214, 39201, 95868, 229644, 569707, 1324730, 3186326, 7664378, 17955006, 42497434, 100710158, 235492595, 549267552, 1288847672, 2990756088, 6958113345, 16148883002, 37286262238, 85880711282, 198840926982, 454980392570

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, A382977.

Cf. A000225, A266964, A048651.

Programs

Magma

n := 31; R := PowerSeriesRing(Rationals(), n+1); f := &*[ (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=31;CoefficientList[Normal@Series[Product[(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 * A382975(k,n-k).

a(n) ~ A048651 * 2^(n-1) * exp(Pi*sqrt(n/3)) / (3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 13 2025