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A201204 Half-convolution of Catalan sequence A000108 with itself. 14
1, 1, 3, 7, 23, 66, 227, 715, 2529, 8398, 30275, 104006, 380162, 1337220, 4939443, 17678835, 65844845, 238819350, 895451117, 3282060210, 12374186318, 45741281820, 173257703723, 644952073662, 2452607696798, 9183676536076, 35042725663002, 131873975875180, 504697422982484, 1907493251046152 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

In general the half-convolution of a sequence {b(n)}_0^infty with itself is defined by chat(n):=sum(b(k)*b(n-k), k=0..floor(n/2)), n>=0. The o.g.f. of the sequence {chat(n)} is obtained from the bisection 2*chat(2*k) - b(k)^2 = c(2*k), k>=0, with the ordinary convolution c(n):=sum(b(k)*b(n-k),k=0..n), n>=0, and 2*chat(2*k+1) = c(2*k+1), k>=0. This leads to the o.g.f.s  for the corresponding even (e) and odd (o) parts:

  2*Chate(x) - B2(x) = Ce(x) and 2*Chato(x) = Co(x), where Chate(x):= sum(chat(2*k)*x^k,k=0..infty), Chato(x):= sum(chat(2*k+1)*x^k,k=0..infty), B2(x) := sum(b(k)^2*x^k, k=0..infty), Ce(x) := sum(c(2*k)*x^k, k=0..infty) and Co(x) := sum(c(2*k+1)*x^k, k=0..infty). Thus Chate(x)=(Ce(x) + B2(x))/2 and Chato(x)=Co(x)/2. Expressing this in terms of C(x), the o.g.f. of {c(n)}, and B2(x) leads to the result: Chat(x)= (C(x) + B2(x^2))/2.

In the Catalan case b(n)=A000108(n), c(n)=b(n+1), C(x)= (cata(x)+1)/x, with the o.g.f. of A000108 cata(x)=(1-sqrt(1-4*x))/(2*x), and B2(x) is found under A001246 to be (-1 + hypergeom([-1/2,-1/2],[1],16*x))/(4*x). This produces the o.g.f. given in the formula section.

This computation was motivated by a question about the o.g.f. of A000992 ("half-Catalan numbers"). Note, however, that this sequence is not the half-convolution of the Catalan numbers presented here.

Apparently the number of hills to the left of or at the midpoint in all Dyck paths of semilength n+1. [David Scambler, Apr 30 2013]

LINKS

Robert Israel, Table of n, a(n) for n = 0..170

FORMULA

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

O.g.f.: G(x)=(catalan(x)-1)/(2*x)+(-1+hypergeom([-1/2,-1/2],[1],16*x^2))/(8*x^2), with the o.g.f. catalan(x) of the Catalan numbers (see also the comment section).

a(n) ~ 2^(2*n+1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 15 2014

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

MAPLE

C:= n -> binomial(2*n, n)/(n+1):

A:= n -> add(C(k)*C(n-k), k=0..floor(n/2));

seq(A(i), i=1..100); # Robert Israel, Jun 06 2014

MATHEMATICA

Table[Sum[CatalanNumber[k]CatalanNumber[n-k], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Harvey P. Dale, Jun 12 2012 *)

Table[CatalanNumber[n + 1]/2 + 2^(2 n + 1) Binomial[1/2, n/2 + 1]^2, {n, 0, 30}] (* Vladimir Reshetnikov, Oct 03 2016 *)

CROSSREFS

A000108, bisection: A201205 and A065097.

Sequence in context: A148690 A148691 A148692 * A106964 A096019 A148693

Adjacent sequences:  A201201 A201202 A201203 * A201205 A201206 A201207

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Jan 02 2012

STATUS

approved

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Last modified May 22 20:59 EDT 2019. Contains 323488 sequences. (Running on oeis4.)