A386432 a(n) = Sum_{k=0..n} n^k * binomial(n,k) * Catalan(k+1).

1, 3, 29, 532, 14849, 562551, 27053749, 1581258225, 108965790593, 8657148898585, 779508506302701, 78480330282178738, 8738801236865140417, 1066555304017996550265, 141604665239501105707269, 20321162053065050407161076, 3134730687100285268294654465, 517309567362171441488395248225

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

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

Mathematica

Table[Sum[(n^k/. 0^0->1)*Binomial[n, k]*CatalanNumber[k+1], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Aug 22 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, n^k*binomial(n, k)*(2*(k+1))!/((k+1)!*(k+2)!));
(Magma) [&+[n^k*Binomial(n, k) * Catalan (k+1): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Aug 22 2025

Crossrefs

Main diagonal of A386408.

Cf. A000108, A338979.

Sequence in context: A380781 A186451 A248828 * A395340 A210827 A092251

Adjacent sequences: A386429 A386430 A386431 * A386433 A386434 A386435

Keyword

nonn

Author

Seiichi Manyama, Aug 20 2025

Status

approved