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 A386835 #15 Aug 06 2025 06:59:30
%S A386835 1,5,30,198,1375,9843,71876,532220,3981645,30023265,227803642,
%T A386835 1737227682,13303481035,102234258623,787997000640,6089345072056,
%U A386835 47161769198809,365986358229645,2845097133606422,22151577531840830,172710278146819959,1348274852150114251
%N A386835 a(n) = Sum_{k=0..n} binomial(2*n+2,k) * binomial(2*n-k-1,n-k).
%F A386835 a(n) = [x^n] (1+x)^(2*n+2)/(1-x)^n.
%F A386835 a(n) = [x^n] 1/((1-x)^3 * (1-2*x)^n).
%F A386835 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(2*n+2,k) * binomial(n-k+2,n-k).
%F A386835 a(n) = Sum_{k=0..n} 2^k * binomial(n+k-1,k) * binomial(n-k+2,n-k).
%F A386835 a(n) ~ 2^(3*n+5) / (27*sqrt(Pi*n)). - _Vaclav Kotesovec_, Aug 06 2025
%t A386835 Table[Sum[Binomial[2*n + 2, k]*Binomial[2*n - k - 1, n - k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Aug 06 2025 *)
%o A386835 (PARI) a(n) = sum(k=0, n, binomial(2*n+2, k)*binomial(2*n-k-1, n-k));
%Y A386835 Cf. A386836, A386837.
%Y A386835 Cf. A116881.
%K A386835 nonn
%O A386835 0,2
%A A386835 _Seiichi Manyama_, Aug 05 2025