A390300 E.g.f. A(x) satisfies A(x) = exp( x/(1-x)^3 * A(x)^2 ).

1, 1, 11, 175, 3993, 121081, 4606963, 211379463, 11368188401, 701655787729, 48898175814171, 3798362200547839, 325457687325948169, 30495421564387422345, 3102169148075839716995, 340490056855504193740951, 40108920625707210968551137, 5047364899444417568755120417

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(n+2*k-1,n-k)/k!.

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

Mathematica

Table[n!*Sum[(2*k+1)^(k-1)*Binomial[n+2*k-1, n-k]/k!, {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Nov 01 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+1)*n+((s-r)*u+t-1)*k+q*u-1, n-k)/k!);
(Magma) [Factorial(n) * &+[(2*k+1)^(k-1)* Binomial(n+2*k-1, n-k) / Factorial(k) : k in [0..n]] : n in [0..25] ]; // Vincenzo Librandi, Nov 01 2025

Crossrefs

Cf. A361065, A362776, A390011.

Sequence in context: A280442 A218330 A386723 * A365034 A196664 A003729

Adjacent sequences: A390297 A390298 A390299 * A390301 A390302 A390303

Keyword

nonn

Author

Seiichi Manyama, Nov 01 2025

Status

approved