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

0,2

Links

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

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A338748 A107945 A279571 * A225294 A124294 A124295

Adjacent sequences: A014327 A014328 A014329 * A014331 A014332 A014333

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Vincenzo Librandi, Feb 27 2014

Status

approved