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 A307947 #20 May 12 2021 03:07:05
%S A307947 1,1,-1,-8,9,201,-89,-8721,-5599,540595,1091551,-43580206,-159753527,
%T A307947 4318835223,24185472039,-506659112324,-3987724324735,68460979242171,
%U A307947 722884820238847,-10431188525128096,-143862807220127799,1761112732849258195,31284729872945906919
%N A307947 Coefficient of x^n in 1/(n+1) * (1 + x - n*x^2)^(n+1).
%C A307947 Also coefficient of x^n in the expansion of 2/(1 - x + sqrt(1 - 2*x + (1+4*n)*x^2)).
%H A307947 Robert Israel, <a href="/A307947/b307947.txt">Table of n, a(n) for n = 0..594</a>
%F A307947 a(n) = Sum_{k=0..floor(n/2)} (-n)^k * binomial(n,k) * binomial(n-k,k)/(k+1) = Sum_{k=0..floor(n/2)} (-n)^k * binomial(n,2*k) * A000108(k).
%F A307947 a(n) = Hypergeometric2F1(1/2 - n/2, -n/2, 2, -4*n). - _Vaclav Kotesovec_, May 12 2021
%p A307947 f:= n -> coeff(1/(n+1)*(1+x-n*x^2)^(n+1),x,n):
%p A307947 map(f, [$0..30]); # _Robert Israel_, May 08 2019
%t A307947 a[0] = 1; a[n_] := Sum[(-n)^k * Binomial[n, 2*k] * CatalanNumber[k], {k, 0, Floor[n/2]}]; Array[a, 23, 0] // Flatten (* _Amiram Eldar_, May 12 2021 *)
%t A307947 Table[Hypergeometric2F1[1/2 - n/2, -n/2, 2, -4*n], {n, 0, 20}] (* _Vaclav Kotesovec_, May 12 2021 *)
%o A307947 (PARI) {a(n) = polcoef((1+x-n*x^2)^(n+1)/(n+1), n)}
%o A307947 (PARI) {a(n) = sum(k=0, n\2, (-n)^k*binomial(n, k)*binomial(n-k, k)/(k+1))}
%o A307947 (PARI) {a(n) = sum(k=0, n\2, (-n)^k*binomial(n, 2*k)*binomial(2*k, k)/(k+1))}
%Y A307947 Cf. A000108, A307862, A307906.
%K A307947 sign
%O A307947 0,4
%A A307947 _Seiichi Manyama_, May 07 2019