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%I #22 Oct 04 2016 06:50:24
%S 1,3,23,227,2529,30275,380162,4939443,65844845,895451117,12374186318,
%T 173257703723,2452607696798,35042725663002,504697422982484,
%U 7319313029400467,106793147620036005,1566546633240722681,23089471526179716182,341774295456352388245
%N Bisection of half-convolution of Catalan sequence A000108; even part.
%C For the definition of the half-convolution of a sequence with itself see a comment to A201204.
%C The odd part of this bisection is found under A065097.
%H Alois P. Heinz, <a href="/A201205/b201205.txt">Table of n, a(n) for n = 0..800</a>
%F a(n) = sum(Catalan(k)*Catalan(2*n-k),k=0..n), n>=0, with Catalan(n)=A000108(n).
%F 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).
%F a(n) = A028364(2n,n) = A067323(2n,n). - _Alois P. Heinz_, Nov 28 2015
%F a(n) = (A000108(2*n+1) + A000108(n)^2)/2. - _Vladimir Reshetnikov_, Oct 03 2016
%p a:= proc(n) option remember; `if`(n<2, 1+2*n,
%p (2*n*(256*n^5-544*n^4+256*n^3+75*n^2-69*n+12)*a(n-1)
%p -(8*(4*n-5))*(4*n-3)*(8*n^2+n-1)*(2*n-3)^2*a(n-2))/
%p ((2*n+1)*n*(8*n^2-15*n+6)*(n+1)^2))
%p end:
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Nov 28 2015
%t Table[(CatalanNumber[2 n + 1] + CatalanNumber[n]^2)/2, {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 03 2016 *)
%Y Cf. A000108, A001246, A028364, A067323, A201204, A065097.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Jan 02 2012
%E Cross-reference corrected by _Robert Israel_, Jun 06 2014
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