OFFSET
0,2
COMMENTS
Binomial transform of A007559.
FORMULA
a(n) = Sum_{k=0..n} binomial(n,k) * A007559(k).
a(n) ~ n! * exp(1/3) * 3^n / (Gamma(1/3) * n^(2/3)). - Vaclav Kotesovec, Aug 14 2021
a(n+2) = (3*n+5)*a(n+1)-3*(n+1)*a(n). - Tani Akinari, Sep 08 2023
MAPLE
g:= proc(n) option remember; `if`(n<2, 1, (3*n-2)*g(n-1)) end:
a:= n-> add(binomial(n, k)*g(k), k=0..n):
seq(a(n), n=0..19); # Alois P. Heinz, Aug 10 2021
MATHEMATICA
nmax = 19; CoefficientList[Series[Exp[x]/(1 - 3 x)^(1/3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] 3^k Pochhammer[1/3, k], {k, 0, n}], {n, 0, 19}]
Table[HypergeometricU[1/3, n + 4/3, 1/3]/3^(1/3), {n, 0, 19}]
PROG
(Maxima) a[n]:=if n<2 then n+1 else (3*n-1)*a[n-1]+3*(1-n)*a[n-2];
makelist(a[n], n, 0, 50); /* Tani Akinari, Sep 08 2023 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Aug 10 2021
STATUS
approved