A201205 Bisection of half-convolution of Catalan sequence A000108; even part.

1, 3, 23, 227, 2529, 30275, 380162, 4939443, 65844845, 895451117, 12374186318, 173257703723, 2452607696798, 35042725663002, 504697422982484, 7319313029400467, 106793147620036005, 1566546633240722681, 23089471526179716182, 341774295456352388245

Offset: 0

Author

Wolfdieter Lang, Jan 02 2012

Keywords

• nonn
• easy

Comments

For the definition of the half-convolution of a sequence with itself see a comment to A201204.

The odd part of this bisection is found under A065097.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..800

Crossrefs

Cf. A000108, A001246, A028364, A067323, A201204, A065097.

Programs

Maple

a:= proc(n) option remember; `if`(n<2, 1+2*n,
      (2*n*(256*n^5-544*n^4+256*n^3+75*n^2-69*n+12)*a(n-1)
       -(8*(4*n-5))*(4*n-3)*(8*n^2+n-1)*(2*n-3)^2*a(n-2))/
      ((2*n+1)*n*(8*n^2-15*n+6)*(n+1)^2))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 28 2015

Mathematica

Table[(CatalanNumber[2 n + 1] + CatalanNumber[n]^2)/2, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 03 2016 *)

Formula

a(n) = sum(Catalan(k)*Catalan(2*n-k),k=0..n), n>=0, with Catalan(n)=A000108(n).

O.g.f: Ge(x)=(catao(x)+cata2(x))/2 with catao(x):= sum(Catalan(2*k+1)*x^k,k=0..infty) = (cata(sqrt(x)) - cata(-sqrt(x)))/(2*x), with the o.g.f. cata(x) of A000108, and cata2(x):=sum(Catalan(n)^2,n=0..infty) given in A001246 as (-1 + hypergeom( [-1/2,-1/2],[1],16*x))/(4*x).

a(n) = A028364(2n,n) = A067323(2n,n). - Alois P. Heinz, Nov 28 2015

a(n) = (A000108(2*n+1) + A000108(n)^2)/2. - Vladimir Reshetnikov, Oct 03 2016

Extensions

Cross-reference corrected by Robert Israel, Jun 06 2014