A382978 -id:A382978 - cp's OEIS frontend

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

Seiichi Manyama, Apr 11 2025

nonn

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

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

Cf. A322199, A382977, A382978.
Cf. A000051, A266964, A284593.
Cf. A079555.

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

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;

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

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

Original entry on oeis.org

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

Seiichi Manyama, Apr 11 2025

nonn

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

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

Cf. A322199, A382976, A382978.

Cf. A000225, A266964.

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

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 *)

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;

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

cp's OEIS Frontend

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

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