A382976 Expansion of Product_{k>=1} (1 + (2^k + 1) * x^k).

1, 3, 5, 24, 44, 129, 384, 897, 2220, 5706, 15268, 35178, 89829, 212982, 526222, 1294263, 3087570, 7300896, 17726100, 41705904, 98782950, 236059794, 551697495, 1293417672, 3033232130, 7081297146, 16430673765, 38347412562, 88762751808, 204970377366, 473719894598

Offset

0,2

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.

Formula

a(n) = Sum_{k=0..n} 2^k * A284593(k,n-k).

a(n) ~ A079555 * 2^(n-1) * exp(Pi*sqrt(n/3)) / (3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Apr 11 2025

Mathematica

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

Prog

(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;

Crossrefs

Cf. A322199, A382977, A382978.

Cf. A000051, A266964, A284593.

Cf. A079555.

Sequence in context: A200948 A275148 A336425 * A230985 A286427 A290509

Adjacent sequences: A382973 A382974 A382975 * A382977 A382978 A382979

Keyword

nonn

Author

Seiichi Manyama, Apr 11 2025

Status

approved