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 A116388 #16 Aug 05 2025 18:04:55
%S A116388 1,1,4,10,29,82,236,681,1975,5745,16757,48982,143442,420721,1235663,
%T A116388 3633453,10695292,31511524,92919758,274203662,809719718,2392579638,
%U A116388 7073684393,20924387460,61925598216,183350728661,543095661673
%N A116388 Expansion of 1/((1+x*(1-M(x)))*sqrt(1-2*x-3*x^2)), M(x) the g.f. of A001006.
%H A116388 G. C. Greubel, <a href="/A116388/b116388.txt">Table of n, a(n) for n = 0..500</a>
%F A116388 G.f.: 2*x/(sqrt(1-2*x-3*x^2)*(sqrt(1-2*x-3*x^2) -1 +2*x +3*x^2)).
%F A116388 a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(n-k,j-k)*C(j,n-k-j).
%F A116388 Conjecture: n*a(n) + 7*(-n+1)*a(n-1) + 2*(4*n-9)*a(n-2) + (25*n-58)*a(n-3) + (-18*n+65)*a(n-4) + (-52*n+199)*a(n-5) + (-31*n+135)*a(n-6) + 6*(-n+5)*a(n-7) = 0. - _R. J. Mathar_, Jun 22 2016
%t A116388 Table[Sum[Sum[Binomial[n-k,j-k]*Binomial[j,n-k-j], {j,0,n-k}], {k,0,Floor[n/2]}], {n,0,30}] (* _G. C. Greubel_, May 23 2019 *)
%o A116388 (PARI) {a(n) = sum(k=0,n\2, sum(j=0,n-k, binomial(n-k, j-k)*binomial(j,n-k-j)))}; \\ _G. C. Greubel_, May 23 2019
%o A116388 (Magma) [(&+[ (&+[Binomial(n-k, j-k)*Binomial(j, n-k-j): j in [0..n-k]]) : k in [0..Floor(n/2)]]): n in [0..30]]; // _G. C. Greubel_, May 23 2019
%o A116388 (Sage) [sum( sum(binomial(n-k, j-k)*binomial(j,n-k-j) for j in (0..n)) for k in (0..floor(n/2))) for n in (0..30)] # _G. C. Greubel_, May 23 2019
%o A116388 (GAP) List([0..30], n-> Sum([0..n], k-> Sum([0..n], j-> Binomial(n-k, j-k)*Binomial(j, n-k-j) ))); # _G. C. Greubel_, May 23 2019
%K A116388 easy,nonn
%O A116388 0,3
%A A116388 _Paul Barry_, Feb 12 2006