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
Links
- Paolo Xausa, Table of n, a(n) for n = 1..1500
Programs
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Mathematica
A034274[n_] := (n^2 + 1)*CatalanNumber[n]/2; Array[A034274, 25] (* Paolo Xausa, Aug 22 2025 *)
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
Extensions
Corrected and extended by N. J. A. Sloane, Apr 21 2000