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A357834
a(n) = Sum_{k=0..floor(n/3)} Stirling1(n,3*k).
3
1, 0, 0, 1, -6, 35, -224, 1603, -12810, 113589, -1109472, 11852841, -137611110, 1726238787, -23277264192, 335861699355, -5164348236138, 84316474011861, -1456893047937600, 26562992204112273, -509679388313669574, 10266675502780006947, -216625348636705401120
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
Cf. A105752.
Sequence in context: A317409 A354324 A213452 * A000399 A081051 A145145
KEYWORD
sign
AUTHOR
Seiichi Manyama, Oct 14 2022
STATUS
approved