A372738 Binomial transform of A369795.

1, 4, 28, 298, 4240, 75394, 1608688, 40045618, 1139279680, 36463487554, 1296712045648, 50724943433938, 2164652356532320, 100072984472662114, 4982304066392196208, 265770533884409878258, 15122101633293034668160, 914210942121577873619074, 58519992421072004957876368, 3954059527570115477197922578

Offset: 0

Author

Prabha Sivaramannair, May 11 2024

Keywords

• nonn

Crossrefs

Cf. A369795, A355408.

Programs

Mathematica

nmax = 20; CoefficientList[Series[E^(2*x)/(1 + E^x - E^(3*x)), {x, 0, nmax}], x]*Range[0, nmax]! (* Vaclav Kotesovec, Jun 01 2024 *)

SageMath

def a(n):
    if n==0:
        return 1
    else:
        return sum([(1-(-1)^j-(-2)^j)*binomial(n,j)*a(n-j) for j in [1,..,n]])
list(a(n) for n in [0,..,20])

Formula

a(n) = Sum_{j=1..n} (1-(-1)^j-(-2)^j)*binomial(n,j)*a(n-j) for n > 0.

a(n) = 2^n + Sum_{j=1..n} (3^j-1)*binomial(n,j)*a(n-j).

a(n) = 1 + Sum_{j=1..n} (2^j-(-1)^j)*binomial(n,j)*a(n-j).

E.g.f.: exp(2*x)/(1 + exp(x) - exp(3*x)). - Vaclav Kotesovec, Jun 01 2024