A137637 - OEIS

%I #6 Nov 25 2023 08:41:07

%S 1,6,32,170,899,4764,25318,134964,721562,3868024,20785035,111931154,

%T 603938905,3264309644,17671408012,95800342628,520022296700,

%U 2826089180652,15374990077568,83727902852188,456370687687082

%N a(n) = Sum_{k=0..n} C(2k+2,k)*C(2k+2,n-k) ; equals row 2 of square array A137634 ; also equals the convolution of A137635 and the self-convolution of A073157.

%H Seiichi Manyama, <a href="/A137637/b137637.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: A(x) = R(x)*G(x)^2, where R(x) = 1/sqrt(1-4*x*(1+x)^2) is the g.f. of A137635 and G(x) = (1-sqrt(1-4*x*(1+x)^2))/(2*x*(1+x)) is the g.f. of A073157.

%o (PARI) {a(n)=sum(k=0,n,binomial(2*k+2,k)*binomial(2*k+2,n-k))} /* Using the g.f.: */ {a(n)=local(R=1/sqrt(1-4*x*(1+x +x*O(x^n))^2), G=(1-sqrt(1-4*x*(1+x)^2+x^2*O(x^n)))/(2*x*(1+x+x*O(x^n)))); polcoeff(R*G^2,n,x)}

%Y Cf. A137634, A137635, A137636, A137638, A073157.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 31 2008