A356363 - OEIS

%I #12 Feb 16 2025 08:34:03

%S 1,0,0,3,24,125,576,3136,24752,242280,2421000,23568743,230156136,

%T 2370756505,26664718080,326641069815,4243004068192,57065900282730,

%U 787656999701016,11193821784313606,165023822310642520,2535785869709189307,40583218821499596176

%N a(n) = Sum_{k=0..floor(n/3)} n^k * Stirling2(n,3*k).

%H Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</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! + ... . a(n) = n! * [x^n] F(n^(1/3) * (exp(x)-1)).

%F a(n) = ( Bell_n(n^(1/3)) + Bell_n(n^(1/3)*w) + Bell_n(n^(1/3)*w^2) )/3, where Bell_n(x) is n-th Bell polynomial.

%o (PARI) a(n) = sum(k=0, n\3, n^k*stirling(n, 3*k, 2));

%o (PARI) a(n) = n!*polcoef(sum(k=0, n\3, n^k*(exp(x+x*O(x^n))-1)^(3*k)/(3*k)!), n);

%o (PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);

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

%Y Cf. A356361, A356362.

%Y Cf. A357782.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Oct 16 2022