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

0,2

Links

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

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

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 *)

Prog

(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

Crossrefs

Cf. A000108, A007317, A092392.

Sequence in context: A356281 A026671 A026876 * A151090 A059278 A151091

Adjacent sequences: A270444 A270445 A270446 * A270448 A270449 A270450

Keyword

nonn

Author

Vladimir Kruchinin, Mar 17 2016

Status

approved