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 A009564 #44 Sep 02 2025 04:06:02
%S A009564 1,-60,15120,-8648640,8821612800,-14079294028800,32382376266240000,
%T A009564 -101421602465863680000,415017197290314178560000,
%U A009564 -2149789081963827444940800000,13750050968240640337841356800000,-106425394494182556214892101632000000,980390734080409707851586040233984000000
%N A009564 E.g.f. sin(x^2)/2, coefficients of x^(4*n + 2).
%H A009564 G. C. Greubel, <a href="/A009564/b009564.txt">Table of n, a(n) for n = 0..182</a>
%F A009564 a(n) = (-1)^n*(2+4*n)!/(2*(1+2*n)!) = (-1)^n*A001813(2*n+1)/2. - _Robert Israel_, Dec 21 2015
%F A009564 From _Amiram Eldar_, Sep 02 2025: (Start)
%F A009564 a(n) = A024343(n)/2.
%F A009564 Sum_{n>=0} 1/a(n) = sqrt(2*Pi) * (cos(1/4) * FresnelC(1/sqrt(2*Pi)) + sin(1/4) * FresnelS(1/sqrt(2*Pi))), where FresnelC(x) and FresnelS(x) are the Fresnel integrals C(x) and S(x), respectively.
%F A009564 Sum_{n>=0} (-1)^n/a(n) = (sqrt(Pi)/2) * (exp(1/4) * erf(1/2) + erfi(1/2) / exp(1/4)). (End)
%p A009564 seq(i!*coeff(series(sin(x^2)/2,x,4*i+4),x,i),i=2..54,4); # _Peter Luschny_, Dec 14 2012
%t A009564 nmax = 12; coes = CoefficientList[ Series[ Sin[x^2]/2, {x, 0, 4*nmax + 2}], x]; a[n_] := coes[[4*n + 3]]*(4*n + 2)!; Table[a[n], {n, 0, nmax}] (* _Jean-François Alcover_, Dec 14 2012 *)
%t A009564 Table[(-1)^n (2 + 4 n)!/(2 (1 + 2 n)!), {n, 0, 25}] (* _Vincenzo Librandi_, Dec 22 2015 *)
%o A009564 (Sage)
%o A009564 def A009564(n):
%o A009564 return falling_factorial(4*n+2,2*n+1)/(2*(-1)^n)
%o A009564 [A009564(n) for n in (0..12)] # _Peter Luschny_, Dec 14 2012
%o A009564 (Magma) [(-1)^n*Factorial(2+4*n)/(2*Factorial(1+2*n)): n in [0..20]]; // _Vincenzo Librandi_, Dec 22 2015
%o A009564 (PARI) a(n) = (-1)^n*(2+4*n)!/(2*(1+2*n)!); \\ _Altug Alkan_, Dec 22 2015
%Y A009564 Cf. A001813, A103639, A024343, A075069.
%K A009564 sign,changed
%O A009564 0,2
%A A009564 _R. H. Hardin_
%E A009564 Extended with signs Mar 1997
%E A009564 Definition corrected and terms a(10)-a(12) from _Peter Luschny_, Dec 14 2012