A357828 - OEIS

%I #16 Oct 15 2022 08:08:29

%S 1,0,0,1,6,35,226,1645,13454,122661,1236018,13656951,164290182,

%T 2138379243,29949509226,449188719525,7183702249542,122039922034485,

%U 2194928052851898,41666342509646127,832547791827455886,17466905709043534107,383908421683657311714

%N a(n) = Sum_{k=0..floor(n/3)} |Stirling1(n,3*k)|.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.

%F 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)).

%F a(n) = ( (1)_n + (w)_n + (w^2)_n )/3, where (x)_n is the Pochhammer symbol.

%o (PARI) a(n) = sum(k=0, n\3, abs(stirling(n, 3*k, 1)));

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N\3, (-log(1-x))^(3*k)/(3*k)!)))

%o (PARI) Pochhammer(x, n) = prod(k=0, n-1, x+k);

%o a(n) = my(w=(-1+sqrt(3)*I)/2); round(Pochhammer(1, n)+Pochhammer(w, n)+Pochhammer(w^2, n))/3;

%Y Cf. A357829, A357830.

%Y Cf. A003703.

%K nonn

%O 0,5

%A _Seiichi Manyama_, Oct 14 2022