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

1, 6, 40, 336, 3456, 42240, 599040, 9676800, 175472640, 3530096640, 78033715200, 1880240947200, 49049763840000, 1377317368627200, 41421544567603200, 1328346084409344000, 45249466617298944000, 1631723190138961920000, 62098722550431350784000, 2487305589722682753024000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100
• Charles Hutton, Circle, A mathematical and philosophical dictionary Vol. 1 (1795), 284-285.
• C. Nicholson, The probability integral for two variables, Biometrika 33 (1943), 59-72.
• Eric Weisstein's World of Mathematics, Normal Distribution Function.
• Index entries for sequences related to factorial numbers.

Crossrefs

From Johannes W. Meijer, Nov 12 2009: (Start)

Appears in A167572.

Equals row sums of A167583. (End)

Cf. A000796, A001147, A009445.

Programs

Haskell

a014481 n = a009445 n `div` a001147 n  -- Reinhard Zumkeller, Dec 03 2011

Magma

[2^n*Factorial(n)*(2*n+1): n in [0..50]]; // Vincenzo Librandi, Apr 25 2011

Mathematica

a[n_]:=2^n*n!*(2*n+1); Array[a,18,0] (* Stefano Spezia, Jan 03 2025 *)

Formula

E.g.f.: (1+2x)/(1-2x)^2.

a(n) = A009445(n) / A001147(n). - Reinhard Zumkeller, Dec 03 2011

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

From Amiram Eldar, Jul 31 2020: (Start)

Sum_{n>=0} 1/a(n) = sqrt(Pi/2) * erfi(1/sqrt(2)).

Sum_{n>=0} (-1)^n/a(n) = sqrt(Pi/2) * erf(1/sqrt(2)). (End)

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

Pi/4 = 1 - Sum_{n>=0} A001147(n)/a(n+1). - Raul Prisacariu, May 20 2025