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A014481 a(n) = 2^n*n!*(2*n+1).

%I A014481 #55 May 21 2025 11:31:19
%S A014481 1,6,40,336,3456,42240,599040,9676800,175472640,3530096640,
%T A014481 78033715200,1880240947200,49049763840000,1377317368627200,
%U A014481 41421544567603200,1328346084409344000,45249466617298944000,1631723190138961920000,62098722550431350784000,2487305589722682753024000
%N A014481 a(n) = 2^n*n!*(2*n+1).
%C A014481 Denominators of expansion of Integral_{t=0..x} exp(-(t^2)/2) dt = sqrt(Pi/2)*erf(x/sqrt(2)) in powers x^(2*n+1), n >= 0. Numerators are (-1)^n. - _Wolfdieter Lang_, Jun 29 2007
%H A014481 Vincenzo Librandi, <a href="/A014481/b014481.txt">Table of n, a(n) for n = 0..100</a>
%H A014481 Charles Hutton, <a href="https://archive.org/details/b28771485_0001/page/284/mode/2up">Circle</a>, A mathematical and philosophical dictionary Vol. 1 (1795), 284-285.
%H A014481 C. Nicholson, <a href="https://www.jstor.org/stable/2333620">The probability integral for two variables</a>, Biometrika 33 (1943), 59-72.
%H A014481 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NormalDistributionFunction.html">Normal Distribution Function</a>.
%H A014481 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.
%F A014481 E.g.f.: (1+2x)/(1-2x)^2.
%F A014481 a(n) = A009445(n) / A001147(n). - _Reinhard Zumkeller_, Dec 03 2011
%F A014481 G.f.: G(0)/(2*x) - 1/x, where G(k)= 1 - 2*x+ 1/(1 - 2*x*(k+1)/(2*x*(k+1) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 24 2013
%F A014481 From _Amiram Eldar_, Jul 31 2020: (Start)
%F A014481 Sum_{n>=0} 1/a(n) = sqrt(Pi/2) * erfi(1/sqrt(2)).
%F A014481 Sum_{n>=0} (-1)^n/a(n) = sqrt(Pi/2) * erf(1/sqrt(2)). (End)
%F A014481 V(h, q) = -h/(q*sqrt(2*Pi)) + Sum_{k>=0} (-1)^k*h*q^(2*k-1)*(q^2+(2*k+1))/(a(k)*sqrt(2*Pi)) = (h/2)*erf(q/sqrt(2)) + h*(exp(-q^2/2) - 1)/(q*sqrt(2*Pi)), where V is Nicholson's V-function. V(h, q) = Integral_{x=0..h} Integral_{y=0..q*x/h} phi(x)*phi(y) dydx, where phi(x) is the standard normal density exp(-x^2/2)/sqrt(2*Pi). - _Thomas Scheuerle_, Jan 21 2025
%F A014481 Pi/4 = 1 - Sum_{n>=0} A001147(n)/a(n+1). - _Raul Prisacariu_, May 20 2025
%t A014481 a[n_]:=2^n*n!*(2*n+1); Array[a,18,0] (* _Stefano Spezia_, Jan 03 2025 *)
%o A014481 (Magma) [2^n*Factorial(n)*(2*n+1): n in [0..50]]; // _Vincenzo Librandi_, Apr 25 2011
%o A014481 (Haskell)
%o A014481 a014481 n = a009445 n `div` a001147 n  -- _Reinhard Zumkeller_, Dec 03 2011
%Y A014481 From _Johannes W. Meijer_, Nov 12 2009: (Start)
%Y A014481 Appears in A167572.
%Y A014481 Equals row sums of A167583. (End)
%Y A014481 Cf. A000796, A001147, A009445.
%K A014481 nonn
%O A014481 0,2
%A A014481 _N. J. A. Sloane_