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 A137638 #6 Jul 01 2023 21:49:21
%S A137638 1,3,15,72,361,1840,9505,49578,260540,1377328,7316373,39020372,
%T A137638 208809544,1120621368,6029023185,32507001876,175604614108,
%U A137638 950233307930,5149691511432,27946158749572,151843410356906,825949622559366
%N A137638 Antidiagonal sums of square array A137634.
%F A137638 G.f.: A(x) = 2*(1+x)/((1+2*x + G(x))*G(x)) where G(x) = sqrt(1 - 4*x*(1+x)^2).
%F A137638 a(n) = Sum_{k=0..n} Sum_{j=0..k} C(n-k+2*j,j)*C(n-k+2*j,k-j).
%F A137638 D-finite with recurrence 2*(n+1)*a(n) +(-3*n-7)*a(n-1) +2*(-17*n+10)*a(n-2) +8*(-7*n+10)*a(n-3) +2*(-18*n+37)*a(n-4) +4*(-2*n+5)*a(n-5)=0. - _R. J. Mathar_, Jun 23 2023
%o A137638 (PARI) {a(n)=sum(k=0,n,sum(j=0,k,binomial(2*j+n-k,j)*binomial(2*j+n-k,k-j)))} /* Using the g.f.: */ {a(n)=local(G=sqrt(1 - 4*x*(1+x)^2 +x*O(x^n))); polcoeff(2*(1+x)/((1+2*x+G)*G),n)}
%Y A137638 Cf. A137634, A137635, A137636, A137637.
%K A137638 nonn
%O A137638 0,2
%A A137638 _Paul D. Hanna_, Jan 31 2008