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 A331795 #27 Aug 24 2025 06:25:48
%S A331795 1,0,-3,40,-515,7056,-102935,1554288,-22775319,265497760,586651461,
%T A331795 -230587852560,13426823564869,-637734419560224,28594259589697425,
%U A331795 -1264238490602458784,56015489395280490385,-2499557487903373341888,112150411888789509887053
%N A331795 a(n) = Sum_{k=0..n} (-n)^k * binomial(n+1,k) * binomial(n+1,k+1).
%H A331795 Seiichi Manyama, <a href="/A331795/b331795.txt">Table of n, a(n) for n = 0..386</a>
%F A331795 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 A331795 a(n) = (n+1) * 2F1(-1 - n, -n; 2; -n), where 2F1 is the hypergeometric function. - _Vaclav Kotesovec_, Jan 26 2020
%F A331795 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 A331795 Flatten[{1, Table[Sum[(-1)^k * n^k * Binomial[n+1,k] * Binomial[n+1,k+1], {k,0,n}], {n,1,20}]}] (* _Vaclav Kotesovec_, Jan 26 2020 *)
%t A331795 Table[(n+1) * Hypergeometric2F1[-1 - n, -n, 2, -n], {n, 0, 20}] (* _Vaclav Kotesovec_, Jan 26 2020 *)
%o A331795 (PARI) a(n) = sum(k=0, n, (-n)^k*binomial(n+1, k)*binomial(n+1, k+1));
%o A331795 (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 A331795 Cf. A331791, A331794.
%K A331795 sign,changed
%O A331795 0,3
%A A331795 _Seiichi Manyama_, Jan 26 2020