A357828 a(n) = Sum_{k=0..floor(n/3)} |Stirling1(n,3*k)|.
1, 0, 0, 1, 6, 35, 226, 1645, 13454, 122661, 1236018, 13656951, 164290182, 2138379243, 29949509226, 449188719525, 7183702249542, 122039922034485, 2194928052851898, 41666342509646127, 832547791827455886, 17466905709043534107, 383908421683657311714
Offset: 0
Keywords
Links
- Eric Weisstein's World of Mathematics, Pochhammer Symbol.
Programs
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PARI
a(n) = sum(k=0, n\3, abs(stirling(n, 3*k, 1)));
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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); round(Pochhammer(1, n)+Pochhammer(w, n)+Pochhammer(w^2, n))/3;
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 + (w)_n + (w^2)_n )/3, where (x)_n is the Pochhammer symbol.
a(n) ~ n!/3. - Vaclav Kotesovec, Jun 10 2025