A167576 The first column of the ED3 array A167572.

1, 5, 23, 167, 1473, 16413, 211479, 3192975, 54010305, 1030249845, 21566327895, 497334999735, 12405876372225, 335591130336525, 9716331072597975, 301633179343890975, 9941514351641143425, 348336799875365041125

Offset: 1

Author

Johannes W. Meijer, Nov 10 2009

Keywords

• easy
• nonn

Comments

Basically a(n) measures the difference between the Euler factorial n! and the Luschny factorial L(n) at half-integer values. For the Luschny factorial see the link. The formula given in the Maple section is a variant of a formula given by Cyril Damamme in A135457. - Peter Luschny, Jul 18 2015

Examples

			G.f. = x + 5*x^2 + 23*x^3 + 167*x^4 + 1473*x^5 + 16413*x^6 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..400
• Peter Luschny, Is the Gamma function misdefined? Hadamard versus Euler - Who found the better Gamma function?

Crossrefs

Equals the first column of the ED3 array A167572.

Equals the first right hand column of A167583.

Other columns are A167577 and A167578.

Cf. A097801 (the 2*(-1)^n*(2*n-5)!! factor).

Cf. A007509 and A025547 (the sum((-1)^(k+n)/(2*k+1), k=0..n-1) factor).

Cf. A024199 and A135457.

Programs

Maple

L := x -> (1+x*(Psi(1-x/2)-Psi(1/2-x/2)))/(-x)!:
a := x -> (L(x-1/2)-(x-1/2)!)*2^(x-1)*sqrt(Pi):
seq(simplify(a(n)),n=1..18); # Peter Luschny, Jul 18 2015
a := proc(n) option remember: if n=1 then 1 else (2*n-1)*a(n-1)+2*(-1)^n*doublefactorial(2*n-5) fi: end: seq(a(n),n=1..18); # Johannes W. Meijer, Jul 20 2015

Mathematica

a[ n_] := If[ n < 1, 0, (2 n - 3)!! ((-1)^n - I (4 n - 2) Sum[ I^k / k, {k, 1, 2 n - 1, 2}])]; (* Michael Somos, Jul 20 2015 *)
a[ n_] := If[ n < 1, 0, (2 n - 3)!! ((-1)^n + (4 n - 2) Sum[ KroneckerSymbol[ -4, k]/ k, {k, 2 n - 1}])]; (* Michael Somos, Jan 31 2019 *)

PARI

{a(n) = if( n<1, 0, prod(k=1, n-1, 2*k - 1) * ((-1)^n - (4*n - 2) * sum(k=1, n, (-1)^k / (2*k - 1))))}; /* Michael Somos, Jul 20 2015 */

Formula

a(n) = (-1)^n*(2*n-3)!!*(1 + (4*n-2)*Sum_{k=0..n-1} (-1)^(k+n)/(2*k+1)).

a(n) = (2*n-1)*a(n-1) + 2*(-1)^n*(2*n-5)!! with a(1) = 1.

a(n) = 4*a(n-1) + (4*n^2 - 16*n + 15)*a(n-2) with a(1) = 1 and a(2) = 5 [Superseeker].

0 = a(n)*a(n+1)*(-440*a(n+2) - 220*a(n+3) + 55*a(n+4)) + a(n)*a(n+2)*(536*a(n+2) - 118*a(n+3) - 4*a(n+4)) + a(n)*a(n+3)*(-4*a(n+3) + a(n+4)) + a(n+1)^2*(-220*a(n+2) - 32*a(n+3) + 8*a(n+4)) + a(n+1)*a(n+2)*(+71*a(n+2) + 4*a(n+3) - 2*a(n+4)) + a(n+2)^2*(-4*a(n+2) + a(n+3)) if n>0. - Michael Somos, Jul 19 2015

a(n) = (-1 + (n-1/2)*LerchPhi(-1,1,n+1/2) + (-n+1/2)*LerchPhi(-1,1,-n+1/2))/(1-2*n)!!. - Johannes W. Meijer, Jul 20 2015

a(n) = A024199(n) + A135457(n). - Cyril Damamme, Jul 22 2015

a(n) = ((-1)^n/(2*n - 1) + Pi/2 - (-1)^n LerchPhi(-1, 1, n + 1/2)) (2*n - 1)!!. - Michael Somos, Jan 31 2019