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 A386940 #22 Aug 12 2025 02:46:11
%S A386940 1,3,13,60,285,1378,6748,33372,166365,834900,4213638,21368724,
%T A386940 108820764,556184580,2851679620,14661848560,75568345821,390330333402,
%U A386940 2020046912260,10472193542100,54373036935910,282704274266040,1471722678992700,7670327017789800,40017679829372700
%N A386940 a(n) = Sum_{k=0..n} binomial(2*k,k) * binomial(2*n-k-1,n-k).
%F A386940 a(n) = [x^n] 1/(sqrt(1-4*x) * (1-x)^n).
%F A386940 G.f.: (1+sqrt(1-4*x))/sqrt( 4 * (1-4*x) * (2*sqrt(1-4*x)-1) ).
%F A386940 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(2*n-1/2,k) * binomial(n-k-1/2,n-k) = Sum_{k=0..n} (3/4)^k * binomial(2*k,k) * binomial(2*n-1/2,n-k).
%F A386940 a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(2*n-1/2,k) * binomial(2*n-k-1,n-k).
%o A386940 (PARI) a(n) = sum(k=0, n, binomial(2*k, k)*binomial(2*n-k-1, n-k));
%Y A386940 Cf. A100193, A384365, A386941.
%Y A386940 Cf. A000984, A293490.
%K A386940 nonn,easy
%O A386940 0,2
%A A386940 _Seiichi Manyama_, Aug 10 2025