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 A024343 #25 Sep 02 2025 04:06:08
%S A024343 2,-120,30240,-17297280,17643225600,-28158588057600,64764752532480000,
%T A024343 -202843204931727360000,830034394580628357120000,
%U A024343 -4299578163927654889881600000,27500101936481280675682713600000
%N A024343 Expansion of e.g.f. sin(x^2) in powers of x^(4*n + 2).
%C A024343 Absolute values are coefficients of expansion of sinh(x^2).
%H A024343 G. C. Greubel, <a href="/A024343/b024343.txt">Table of n, a(n) for n = 0..175</a>
%F A024343 a(n) = (-1)^n * (4*n+2)! / (2*n+1)!.
%F A024343 E.g.f.: [x^(4*n+2)] sin(x^2)
%F A024343 a(n) = 2 * A009564(n). - _Sean A. Irvine_, Jul 01 2019
%F A024343 From _Amiram Eldar_, Sep 02 2025: (Start)
%F A024343 Sum_{n>=0} 1/a(n) = sqrt(Pi/2) * (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 A024343 Sum_{n>=0} (-1)^n/a(n) = (sqrt(Pi)/4) * (exp(1/4) * erf(1/2) + erfi(1/2) / exp(1/4)). (End)
%t A024343 Table[(-1)^n*(2*n+1)!*Binomial[4*n+2, 2*n+1], {n,0,30}] (* _G. C. Greubel_, Jan 29 2022 *)
%o A024343 (PARI) a(n)=polcoeff(serlaplace(sin(x^2)),4*n+2)
%o A024343 (PARI) a(n)=(-1)^n*(4*n+2)!/(2*n+1)!
%o A024343 (Sage) f=factorial; [(-1)^n*f(4*n+2)/f(2*n+1) for n in (0..30)] # _G. C. Greubel_, Jan 29 2022
%o A024343 (Magma) F:=Factorial;; [(-1)^n*F(4*n+2)/F(2*n+1) : n in [0..30]]; // _G. C. Greubel_, Jan 29 2022
%Y A024343 Bisection of A001813.
%Y A024343 Cf. A009564.
%K A024343 sign,changed
%O A024343 0,1
%A A024343 _R. H. Hardin_
%E A024343 Extended with signs by _Olivier Gérard_, Mar 15 1997
%E A024343 Edited by _Ralf Stephan_, Mar 25 2004
%E A024343 Name edited by _Michel Marcus_, Jul 01 2019