A372738 - OEIS

%I #17 Jun 28 2024 22:26:08

%S 1,4,28,298,4240,75394,1608688,40045618,1139279680,36463487554,

%T 1296712045648,50724943433938,2164652356532320,100072984472662114,

%U 4982304066392196208,265770533884409878258,15122101633293034668160,914210942121577873619074,58519992421072004957876368,3954059527570115477197922578

%N Binomial transform of A369795.

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

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

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

%F E.g.f.: exp(2*x)/(1 + exp(x) - exp(3*x)). - _Vaclav Kotesovec_, Jun 01 2024

%t 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 *)

%o (SageMath)

%o def a(n):

%o     if n==0:

%o         return 1

%o     else:

%o         return sum([(1-(-1)^j-(-2)^j)*binomial(n,j)*a(n-j) for j in [1,..,n]])

%o list(a(n) for n in [0,..,20])

%Y Cf. A369795, A355408.

%K nonn

%O 0,2

%A _Prabha Sivaramannair_, May 11 2024