A034274 - OEIS

%I #28 Sep 04 2025 01:54:37

%S 1,5,25,119,546,2442,10725,46475,199342,848198,3585946,15080870,

%T 63146500,263432340,1095517485,4543460595,18798494550,77616288750,

%U 319874637390,1316106144210,5407045011420,22184521682700,90910797617250,372137346502974,1521789223654476,6217349014923452

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

%H Paolo Xausa, <a href="/A034274/b034274.txt">Table of n, a(n) for n = 1..1500</a>

%F From _Peter Bala_, Aug 19 2025: (Start)

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

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

%F 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).

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

%F a(n) ~ 2^(2*n-1) * sqrt(n/Pi). - _Amiram Eldar_, Sep 04 2025

%t A034274[n_] := (n^2 + 1)*CatalanNumber[n]/2; Array[A034274, 25] (* _Paolo Xausa_, Aug 22 2025 *)

%Y Cf. A000984, A000108, A034261, A180266.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_

%E Corrected and extended by _N. J. A. Sloane_, Apr 21 2000