A270447 Binomial transform(2) of Catalan numbers.

1, 3, 11, 43, 174, 721, 3044, 13059, 56837, 250690, 1119612, 5059561, 23119628, 106753404, 497762380, 2342096579, 11113027686, 53138757319, 255892224332, 1240217043450, 6046131132030, 29631889507380, 145923474439800, 721733515299225, 3583733352377724

Offset: 0

Author

Vladimir Kruchinin, Mar 17 2016

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000108, A007317, A092392.

Programs

Mathematica

Table[Sum[Binomial[2*k,k]/(k+1) * Binomial[2*n-k,n], {k,0,n}], {n,0,25}] (* Vaclav Kotesovec, Mar 17 2016 *)
a[n_] := ((2 n + 1) Binomial[2 n, n] (1 - Hypergeometric2F1[-1/2, -n - 1, -2 n - 1, 4]))/(2 (n + 1));
Table[a[n], {n, 0, 24}] (* Peter Luschny, May 30 2022 *)

Maxima

a(n):=sum((binomial(2*k,k)*binomial(2*n-k,n))/(k+1),k,0,n);

PARI

a(n) = sum(i=0, n, (binomial(2*i, i)*binomial(2*n-i, n))/(i+1)); \\ Altug Alkan, Mar 17 2016

Formula

a(n) = Sum_{k=0..n} (T(n,k)*C(k)), where C(k) is Catalan numbers (A000108), T(n,k) - triangle of A092392.

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

G.f.: C(C(x))*(1-C(x))^2/(((1-C(x))^2)-x)/x, where C(x)=(1-sqrt(1-4*x))/2.

Recurrence: 3*(n-1)*n*(n+1)*(2*n - 3)*a(n) = 16*(n-1)*n*(5*n^2 - 10*n + 3)*a(n-1) - 16*(n-1)*(2*n - 1)*(11*n^2 - 33*n + 24)*a(n-2) + 8*(2*n - 3)*(2*n - 1)*(4*n - 9)*(4*n - 7)*a(n-3). - Vaclav Kotesovec, Mar 17 2016

a(n) ~ 2^(4*n + 1/2) / (sqrt(Pi) * 3^(n - 1/2) * n^(3/2)). - Vaclav Kotesovec, Mar 17 2016

a(n) = [x^n] (1 - sqrt(1 - 4*x))/(2*x*(1 - x)^(n+1)). - Ilya Gutkovskiy, Nov 01 2017