A034274 a(n)=f(n,n-1) where f is given in A034261.

1, 5, 25, 119, 546, 2442, 10725, 46475, 199342, 848198, 3585946, 15080870, 63146500, 263432340, 1095517485, 4543460595, 18798494550, 77616288750, 319874637390, 1316106144210, 5407045011420, 22184521682700, 90910797617250, 372137346502974, 1521789223654476, 6217349014923452

Offset

1,2

Links

Paolo Xausa, Table of n, a(n) for n = 1..1500

Formula

From Peter Bala, Aug 19 2025: (Start)

a(n) = (n^2 + 1)/2 * A000108(n).

a(n) = (1/2) * A180266(n+1).

a(n) = Sum_{k = 1..n} k^2/(n+k-1) * binomial(n+k-1, k). Cf. Sum_{k = 1..n} k/(n+k-1) * binomial(n+k-1, k) = 1/2 * binomial(2*n, n) = 1/2 * A000984(n).

a(n) = 2*(n^2 + 1)*(2*n - 1)/((n + 1)*(n^2 - 2*n + 2)) * a(n-1) with a(1) = 1. (End)

a(n) ~ 2^(2*n-1) * sqrt(n/Pi). - Amiram Eldar, Sep 04 2025

Mathematica

A034274[n_] := (n^2 + 1)*CatalanNumber[n]/2; Array[A034274, 25] (* Paolo Xausa, Aug 22 2025 *)

Crossrefs

Cf. A000984, A000108, A034261, A180266.

Sequence in context: A357598 A238808 A383993 * A110212 A344097 A268940

Adjacent sequences: A034271 A034272 A034273 * A034275 A034276 A034277

Keyword

nonn,easy

Author

Clark Kimberling

Extensions

Corrected and extended by N. J. A. Sloane, Apr 21 2000

Status

approved