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

0,2

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

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

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 *)

Crossrefs

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

Sequence in context: A328808 A206763 A306154 * A068954 A068955 A151393

Adjacent sequences: A201202 A201203 A201204 * A201206 A201207 A201208

Keyword

nonn,easy

Author

Wolfdieter Lang, Jan 02 2012

Extensions

Cross-reference corrected by Robert Israel, Jun 06 2014

Status

approved