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 A331794 #30 Aug 24 2025 06:25:00
%S A331794 1,4,33,400,6285,120456,2714173,70129984,2040655401,65956468600,
%T A331794 2342384363561,90607200956064,3789863084012629,170370561866229648,
%U A331794 8188781210421259365,418938023982360898816,22724122083014879989905,1302374806940392958470104
%N A331794 a(n) = Sum_{k=0..n} n^k * binomial(n+1,k) * binomial(n+1,k+1).
%H A331794 Seiichi Manyama, <a href="/A331794/b331794.txt">Table of n, a(n) for n = 0..380</a>
%F A331794 a(n) = [x^n] 2/(1 - 2*(n+1)*x + ((n-1)*x)^2 + (1 - (n+1)*x) * sqrt(1 - 2*(n+1)*x + ((n-1)*x)^2)).
%F A331794 a(n) = (n+1) * 2F1(-1 - n, -n; 2; n), where 2F1 is the hypergeometric function. - _Vaclav Kotesovec_, Jan 26 2020
%F A331794 a(n) = Sum_{k=0..floor(n/2)} n^k * (n+1)^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k). - _Seiichi Manyama_, Aug 24 2025
%t A331794 Flatten[{1, Table[Sum[n^k * Binomial[n+1,k] * Binomial[n+1,k+1], {k,0,n}], {n,1,20}]}] (* _Vaclav Kotesovec_, Jan 26 2020 *)
%t A331794 Table[(n+1) * Hypergeometric2F1[-1 - n, -n, 2, n], {n, 0, 20}] (* _Vaclav Kotesovec_, Jan 26 2020 *)
%o A331794 (PARI) a(n) = sum(k=0, n, n^k*binomial(n+1, k)*binomial(n+1, k+1));
%o A331794 (PARI) a(n) = polcoef(2/(1-2*(n+1)*x+((n-1)*x)^2+(1-(n+1)*x)*sqrt(1-2*(n+1)*x+((n-1)*x)^2)), n);
%Y A331794 Cf. A331791, A331795.
%K A331794 nonn,changed
%O A331794 0,2
%A A331794 _Seiichi Manyama_, Jan 26 2020