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 A380612 #40 Sep 02 2025 04:07:06
%S A380612 1,3,-15,315,-14175,1091475,-127702575,21070924875,-4656674397375,
%T A380612 1327152203251875,-473793336560919375,207047688077121766875,
%U A380612 -108700036240488927609375,67502722505343624045421875,-48939473816374127432930859375,40962339584305144661363129296875
%N A380612 a(n) = (-1)^n*Product_{k=1..n} (2*k + 1)*(2*k - 3).
%F A380612 a(n) = Product_{k=1..n}(2 - (2*k - 1))*Product_{k=1..n}(2 + (2*k - 1)).
%F A380612 a(n) = -(-1)^n*A001147(n-1)*A001147(n+1), for n > 0.
%F A380612 a(n) = Sum_{k=0..n} A380570(n, k).
%F A380612 a(n) = (-(-1)^n*2^(-2*n - 1)*(2*n + 2)!*binomial(2*n, n))/((n + 1)*(2*n - 1)).
%F A380612 a(n) = -(-4)^n*Gamma(-1/2 + n)*Gamma(3/2 + n)/Pi.
%F A380612 a(n) = -(-1)^n*(2*n+1)!*Gamma(n - 1/2)/(2*sqrt(Pi)*n!).
%F A380612 a(n) = Integral_{u=0..1} Integral_{v=0..1} log(1/u)^(-3/2+n)*log(1/v)^(1/2+n) dv du.
%F A380612 Sum_{k>=0} x^(2*k-1)/a(k) = -(Pi/2)*H_v(x), where H_v is the Struve Function at v = -2.
%F A380612 Sum_{k>=0} x^(2*k-1)/abs(a(k)) = (Pi/2)*L_v(x), where L_v is the modified Struve Function at v = -2.
%F A380612 a(n) = (n!)^2 [x^n] hypergeometric([-1/2, 3/2], [1], -4*x).
%F A380612 a(n) = -(-4)^n*Pochhammer(1/2, n - 1)*Pochhammer(1/2, n + 1). - _Peter Luschny_, Jan 29 2025
%F A380612 a(n) ~ (-1)^(n+1) * 2^(2*n+1) * (n/e)^(2*n). - _Amiram Eldar_, Sep 02 2025
%p A380612 gf := hypergeom([-1/2, 3/2], [1], -4*x): ser := series(gf, x, 17):
%p A380612 seq((n!)^2*coeff(ser,x,n), n = 0..15); # _Peter Luschny_, Jan 29 2025
%t A380612 Table[(-1)^n*Product[(2k+1)*(2k-3),{k,n}],{n,0,15}] (* _James C. McMahon_, Feb 02 2025 *)
%o A380612 (PARI) a(n) = Vec(hypergeom([-1/2, 3/2], 1, -4*x)+O(x^(n+1)))[n+1]*(n!)^2
%o A380612 (PARI) a(n) = prod(k=1,n,(2*k+1)*(2*k-3))*(-1)^n
%o A380612 (SageMath)
%o A380612 def a(n): return -(-4)**n*rising_factorial(1/2, n-1)*rising_factorial(1/2, n+1)
%o A380612 print([a(n) for n in range(16)]) # _Peter Luschny_, Jan 29 2025
%Y A380612 Cf. A001147, A380570.
%K A380612 sign,easy,changed
%O A380612 0,2
%A A380612 _Thomas Scheuerle_, Jan 28 2025