A355218 a(n) = Sum_{k>=1} (3*k - 1)^n / 2^k.

1, 5, 43, 557, 9643, 208685, 5419243, 164184557, 5684837803, 221440158125, 9584118542443, 456289689634157, 23698327407870763, 1333388917719691565, 80794290325166308843, 5245268489291712773357, 363231496206350038884523, 26725646191850556128889005, 2082075690178933613292014443

Offset: 0

Author

Ilya Gutkovskiy, Jun 24 2022

Keywords

• nonn

Crossrefs

Cf. A000629, A000670, A007047, A080253, A151919, A328182, A355219, A355220.

Programs

Mathematica

nmax = 18; CoefficientList[Series[Exp[2 x]/(2 - Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2^n + Sum[Binomial[n, k] 3^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

Formula

E.g.f.: exp(2*x) / (2 - exp(3*x)).

a(0) = 1; a(n) = 2^n + Sum_{k=1..n} binomial(n,k) * 3^k * a(n-k).

a(n) = Sum_{k=0..n} binomial(n,k) * 2^(n-k) * 3^k * A000670(k).

a(n) ~ n! * 3^n / (2^(1/3) * log(2)^(n+1)). - Vaclav Kotesovec, Jun 24 2022