This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A137636 #9 Nov 25 2023 08:36:34
%S A137636 1,4,19,94,474,2431,12609,65972,347524,1840680,9792986,52296799,
%T A137636 280163091,1504969409,8103433329,43722788132,236340999038,
%U A137636 1279602656590,6938126362948,37668424608552,204751452911832,1114151447523038
%N A137636 a(n) = Sum_{k=0..n} C(2k+1,k)*C(2k+1,n-k) ; equals row 1 of square array A137634; also equals the convolution of A137635 and A073157.
%H A137636 Seiichi Manyama, <a href="/A137636/b137636.txt">Table of n, a(n) for n = 0..1000</a>
%F A137636 G.f.: A(x) = R(x)*G(x), where R(x) = 1/sqrt(1-4x(1+x)^2) is the g.f. of A137635 and G(x) = (1-sqrt(1-4x(1+x)^2))/(2x(1+x)) is the g.f. of A073157.
%F A137636 D-finite with recurrence (n+1)*a(n) +(-3*n-1)*a(n-1) +2*(-6*n-1)*a(n-2) +2*(-6*n+1)*a(n-3) +2*(-2*n+1)*a(n-4)=0. - _R. J. Mathar_, Jun 23 2023
%F A137636 a(n) ~ sqrt((172 + (86*(78905 - 519*sqrt(129)))^(1/3) + (86*(78905 + 519*sqrt(129)))^(1/3))/129) * ((4 + (262 - 6*sqrt(129))^(1/3) + (2*(131 + 3*sqrt(129)))^(1/3))/3)^n / sqrt(Pi*n). - _Vaclav Kotesovec_, Nov 25 2023
%o A137636 (PARI) {a(n)=sum(k=0,n,binomial(2*k+1,k)*binomial(2*k+1,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,n,x)}
%Y A137636 Cf. A137634, A137635, A137637, A137638, A073157.
%K A137636 nonn
%O A137636 0,2
%A A137636 _Paul D. Hanna_, Jan 31 2008