A357829 a(n) = Sum_{k=0..floor((n-1)/3)} |Stirling1(n,3*k+1)|.

0, 1, 1, 2, 7, 34, 205, 1456, 11837, 108150, 1096011, 12196128, 147814359, 1938062490, 27333191613, 412614191808, 6638401596645, 113398127795862, 2049808094564139, 39091473755006400, 784404343854767727, 16520634668922810426, 364400233756422553053

Offset

0,4

Links

Table of n, a(n) for n=0..22.

Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Formula

Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + w^2*exp(w*x) + w*exp(w^2*x))/3 = x + x^4/4! + x^7/7! + ... . Then the e.g.f. for the sequence is F(-log(1-x)).

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

Prog

(PARI) a(n) = sum(k=0, (n-1)\3, abs(stirling(n, 3*k+1, 1)));
(PARI) my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=0, N\3, (-log(1-x))^(3*k+1)/(3*k+1)!))))
(PARI) Pochhammer(x, n) = prod(k=0, n-1, x+k);
a(n) = my(w=(-1+sqrt(3)*I)/2); round(Pochhammer(1, n)+w^2*Pochhammer(w, n)+w*Pochhammer(w^2, n))/3;

Crossrefs

Cf. A357828, A357830.

Sequence in context: A145345 A212027 A056543 * A075834 A011800 A112916

Adjacent sequences: A357826 A357827 A357828 * A357830 A357831 A357832

Keyword

nonn

Author

Seiichi Manyama, Oct 14 2022

Status

approved