A390057 E.g.f. A(x) satisfies A(x) = exp( x * (1+x)^3 * A(x)^2 ).

1, 1, 11, 157, 3417, 99661, 3653323, 161681913, 8391344657, 499955684761, 33639020702091, 2523130141439029, 208768016202204649, 18890958253984401573, 1855889522744334084491, 196730385503728744962961, 22381974409910566684128417, 2720322634728180274462729777

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Formula

a(n) = n! * Sum_{k=0..n} (2*k+1)^(k-1) * binomial(3*k,n-k)/k!.

E.g.f.: exp( -LambertW(-2*x * (1+x)^3)/2 ).

Mathematica

Table[n!*Sum[(2*k+1)^(k-1)*Binomial[3*k, n-k]/k!, {k, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Oct 27 2025 *)

Prog

(PARI) a(n, q=1, r=0, s=2, t=3, u=0) = q*n!*sum(k=0, n, (r*n+(s-r)*k+q)^(k-1)*binomial(r*u*n+((s-r)*u+t)*k+q*u, n-k)/k!);
(Magma) [Factorial(n) * &+[(2*k+1)^(k-1) * Binomial(3*k, n-k) / Factorial(k) : k in [0..n]] : n in [0..25] ]; // Vincenzo Librandi, Oct 27 2025

Crossrefs

Cf. A052750, A362773, A362774, A390063, A390064.

Cf. A376145.

Sequence in context: A156933 A242008 A158512 * A379687 A348151 A324430

Adjacent sequences: A390054 A390055 A390056 * A390058 A390059 A390060

Keyword

nonn

Author

Seiichi Manyama, Oct 23 2025

Status

approved