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 A116389 #8 Sep 08 2022 08:45:24
%S A116389 1,0,1,2,2,1,0,4,4,1,6,10,10,6,1,0,16,28,20,8,1,20,44,62,62,34,10,1,0,
%T A116389 64,152,168,120,52,12,1,70,186,328,436,374,210,74,14,1,0,256,748,1084,
%U A116389 1072,736,340,100,16,1,252,772,1606,2598,2924,2332,1326,518,130,18,1
%N A116389 Riordan array (1/sqrt(1-4*x^2), (1+x)/sqrt(1-4*x^2) -1).
%H A116389 G. C. Greubel, <a href="/A116389/b116389.txt">Rows n = 0..100 of triangle, flattened</a>
%F A116389 T(n,k) = Sum_{j=0..k} (-1)^(k-j)*C(k,j) * Sum_{i=0..floor(n/2)} 4^i * C(i+(j-1)/2, i)*C(j,n-2*i).
%e A116389 Triangle begins:
%e A116389 1;
%e A116389 0, 1;
%e A116389 2, 2, 1;
%e A116389 0, 4, 4, 1;
%e A116389 6, 10, 10, 6, 1;
%e A116389 0, 16, 28, 20, 8, 1;
%e A116389 20, 44, 62, 62, 34, 10, 1;
%t A116389 T[n_,k_]:= Sum[(-1)^(k-j)*Binomial[k,j]*Sum[4^r*Binomial[r+(j-1)/2, r]* Binomial[j, n-2*r], {r,0,Floor[n/2]}], {j,0,k}]; Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 23 2019 *)
%o A116389 (PARI) {T(n,k) = sum(j=0,k, sum(r=0,floor(n/2), (-1)^(k-j)*4^r* binomial(k,j)*binomial(r+(j-1)/2, r)*binomial(j, n-2*r) ))}; \\ _G. C. Greubel_, May 23 2019
%o A116389 (Magma) [[(&+[ (&+[ Round((-1)^(k-j)*4^r*Binomial(k,j)*Binomial(j, n-2*r)*Gamma(r+(j+1)/2)/(Factorial(r)*Gamma((j+1)/2))) : r in [0..Floor(n/2)]]) : j in [0..k]]) : k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, May 23 2019
%o A116389 (Sage) [[sum( sum( (-1)^(k-j)*4^r* binomial(k,j)*binomial(r+(j-1)/2, r)*binomial(j, n-2*r) for r in (0..floor(n/2))) for j in (0..k)) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, May 23 2019
%Y A116389 Row sums are A116390. Diagonal sums are A116391.
%Y A116389 Product of A007318 and this sequence is A116392.
%K A116389 easy,nonn,tabl
%O A116389 0,4
%A A116389 _Paul Barry_, Feb 12 2006