A014330 Exponential convolution of Catalan numbers with themselves.

1, 2, 6, 22, 92, 424, 2108, 11134, 61748, 356296, 2123720, 13002840, 81417520, 519550880, 3369559864, 22161337742, 147544048324, 992923683912, 6746101933304, 46226667046360, 319199694771696, 2219445498261152, 15529758665102416, 109291258152550712

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000108, A014333, A053175, A126869, A138364.

Programs

Magma

A014330:= func< n | (&+[Binomial(n,k)*Catalan(k)*Catalan(n-k): k in [0..n]]) >;
[A014330(n): n in [0..40]]; // G. C. Greubel, Jan 06 2023

Mathematica

Table[Sum[Binomial[n, k]*Binomial[2*k, k]/(k+1)*Binomial[2*n-2*k, n-k]/(n-k+1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Feb 25 2014 *)

PARI

A014330(n)=sum(k=0,n,binomial(n,k)*A000108(k)*A000108(n-k))  \\ M. F. Hasler, Jan 13 2012

SageMath

def c(n): return catalan_number(n)
def A014330(n): return sum( binomial(n,k)*c(k)*c(n-k) for k in range(n+1))
[A014330(n) for n in range(41)] # G. C. Greubel, Jan 06 2023

Formula

From Vladeta Jovovic, Jan 01 2004: (Start)

E.g.f.: exp(4*x)*(BesselI(0, 2*x) - BesselI(1, 2*x))^2.

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

a(n) = 4^n*Sum_{k=0..n} (-4)^(-k)*binomial(n, k)*binomial(k, floor(k/2))*binomial(k+1, floor((k+1)/2)).

a(n) = binomial(2*n, n)/(n+1)*hypergeometric3F2([-n-1, -n, 1/2], [2, 1/2-n], -1). (End)

(n + 1)*(n + 2)*a(n) = 4*(3*n^2 + n - 1)*a(n - 1) - 32*(n - 1)^2*a(n - 2). - Vladeta Jovovic, Jul 15 2004

a(n) = Sum_{k=0..n} binomial(n,k)*A000108(k)*A000108(n-k). - Philippe Deléham, Aug 23 2006

a(n) = (4*A053175(n) - A053175(n+1)/4) / ((n+2)*2^n). - Mark van Hoeij, Jul 02 2010

G.f.: (1-6*x)*hypergeometric2F1([1/2, 1/2],[2],16*x^2/(4*x-1)^2)/(2*x*(4*x-1)) - x*(8*x-1)*hypergeometric2F1([3/2, 3/2],[3],16*x^2/(4*x-1)^2)/(4*x-1)^3 + 1/(2*x). - Mark van Hoeij, Oct 25 2011

E.g.f.: hypergeometric1F1([1/2],[2],4*x)^2, coinciding with the above given e.g.f. - Wolfdieter Lang, Jan 13 2012

a(n) ~ 8^(n+1) / (Pi*n^3). - Vaclav Kotesovec, Feb 25 2014

Extensions

More terms from Vincenzo Librandi, Feb 27 2014