A357829 - OEIS

%I #12 Oct 15 2022 08:08:25

%S 0,1,1,2,7,34,205,1456,11837,108150,1096011,12196128,147814359,

%T 1938062490,27333191613,412614191808,6638401596645,113398127795862,

%U 2049808094564139,39091473755006400,784404343854767727,16520634668922810426,364400233756422553053

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

%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) + 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)).

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

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

%o (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)!))))

%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)+w^2*Pochhammer(w, n)+w*Pochhammer(w^2, n))/3;

%Y Cf. A357828, A357830.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Oct 14 2022