A204452 A014330 - A203577. Difference between the exponential convolution of A000108 (Catalan) with itself and the corresponding exponential half-convolution.

0, 1, 2, 11, 34, 212, 804, 5567, 24014, 178148, 839596, 6501420, 32658872, 259775440, 1368965576, 11080668871, 60613092662, 496461841956, 2798385807012, 23113333523180, 133539494791000, 1109722749130576, 6545965568001272

Offset

0,3

Comments

For the exponential (also known as binomial) half-convolution of the Catalan sequence A000108 with itself see A203577.

Links

Table of n, a(n) for n=0..22.

Formula

a(n) = sum(binomial(n,k)*C(k)*C(n-k),k=floor(n/2)+1..n), n>=0, with C(n)=A000108(n), the Catalan numbers.

E.g.f.: (C(x)^2 - C2(x^2))/2 with the e.g.f. C(x) of A000108, and the o.g.f. C2(x) of the sequence {(C(n)/n!)^2}. Compare this with the e.g.f. of A203577.

C(x) = hypergeom([1/2],[2],4*x) (see the e.g.f. of A000108 for the version involving BesselI functions), and

C2(x) = hypergeom([1/2,1/2],[1,2,2],16*x).

Example

With A000108 = {1, 1, 2, 5, 14, 42,...}
  a(4) = 4*5*1 + 1*14*1 = 34.
  a(5) = 10*5*2 + 5*14*1 + 1*42*1 = 212.

Maple

A204452 := proc(n)
    add( binomial(n, k)*A000108(k)*A000108(n-k), k=floor(n/2)+1..n) ;
end proc:
seq(A204452(n), n=0..50) ; # R. J. Mathar, Jul 27 2022

Crossrefs

Cf. A000108, A014330, A203577.

Sequence in context: A027223 A027229 A026946 * A337640 A041389 A205342

Adjacent sequences: A204449 A204450 A204451 * A204453 A204454 A204455

Keyword

nonn,easy

Author

Wolfdieter Lang, Jan 16 2012

Status

approved