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 A299507 #14 Jun 27 2025 18:15:48
%S A299507 1,3,33,387,4737,59523,761121,9854211,128772609,1694927619,
%T A299507 22437369633,298419470979,3984500221569,53376363001731,
%U A299507 717044895641121,9656091923587587,130310873022310401,1761872309456567811,23861153881099854369,323634591584064809859
%N A299507 a(n) = (-1)^n*hypergeom([-n, n], [1], 4).
%F A299507 From _Vaclav Kotesovec_, Jul 05 2018: (Start)
%F A299507 Recurrence: n*(2*n-3)*a(n) = 2*(14*n^2 - 28*n + 11)*a(n-1) - (n-2)*(2*n-1)*a(n-2).
%F A299507 a(n) ~ 2^(-3/2) * 3^(1/4) * (7 + 4*sqrt(3))^n / sqrt(Pi*n). (End)
%F A299507 From _Peter Bala_, Apr 18 2024: (Start)
%F A299507 a(n) = Sum_{k = 0..n} binomial(n, k)*binomial(n+k-1, k-1)*3^k = R(n, 3) for n >= 1, where R(n, x) denotes the n-th row polynomial of A253283.
%F A299507 a(n) = 3*n* hypergeom([1 - n, n + 1], [2], -3) for n >= 1.
%F A299507 a(n) = (1/2)*(LegendreP(n, 7) - LegendreP(n-1, 7)) for n >= 1.
%F A299507 a(n) = [x^n] ( (1 - x)/(1 - 4*x) )^n.
%F A299507 It follows that the Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
%F A299507 G.f.: (sqrt(x^2 - 14*x + 1) - x + 1)/(2*sqrt(x^2 - 14*x + 1)) = 1 + 3*x + 33*x^2 + 387*x^3 + .... (End)
%p A299507 seq(simplify( (-1)^n*hypergeom([-n, n], [1], 4)), n = 0..20); # _Peter Bala_, Apr 18 2024
%t A299507 a[n_] := (-1)^n Hypergeometric2F1[-n, n, 1, 4]; Table[a[n], {n, 0, 19}]
%Y A299507 Cf. A299864, A245926, A084768, A245927, A253283.
%Y A299507 Cf. A300945, A300946.
%K A299507 nonn
%O A299507 0,2
%A A299507 _Peter Luschny_, Mar 16 2018