OFFSET
0,5
LINKS
Eric Weisstein's World of Mathematics, Pochhammer Symbol.
FORMULA
Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + exp(w*x) + exp(w^2*x))/3 = 1 + x^3/3! + x^6/6! + ... . Then the e.g.f. for the sequence is F(log(1+x)).
a(n) = (-1)^n * ( (-1)_n + (-w)_n + (-w^2)_n )/3, where (x)_n is the Pochhammer symbol.
PROG
(PARI) a(n) = sum(k=0, n\3, stirling(n, 3*k, 1));
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N\3, log(1+x)^(3*k)/(3*k)!)))
(PARI) Pochhammer(x, n) = prod(k=0, n-1, x+k);
a(n) = my(w=(-1+sqrt(3)*I)/2); (-1)^n*round(Pochhammer(-1, n)+Pochhammer(-w, n)+Pochhammer(-w^2, n))/3;
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Oct 14 2022
STATUS
approved