A059371 a(n) = (n-1)! + ((n+1)/2)*a(n-1), a(1)=0.

1, 4, 16, 72, 372, 2208, 14976, 115200, 996480, 9607680, 102366720, 1195568640, 15193785600, 208728576000, 3081867264000, 48659595264000, 817953583104000, 14581909536768000, 274755150544896000, 5455208664170496000, 113825841809670144000

Offset: 2

Author

N. J. A. Sloane, Jan 28 2001

Keywords

• nonn
• easy

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 171, #34.

Links

• Harry J. Smith, Table of n, a(n) for n = 2..200

Crossrefs

Second diagonal of triangle in A059369.

Programs

Maple

series(hypergeom([1,2],[],x)^2, x=0, 30);  # Mark van Hoeij, Apr 20 2013

Mathematica

Rest[Rest[CoefficientList[Series[(x^2-2*x-2*Log[1-x])/(x-2)^2, {x, 0, 20}], x]* Range[0, 20]!]] (* Vaclav Kotesovec, Aug 11 2013 *)
Table[-2 n! - 2 (n + 1)! Re[LerchPhi[2, 1, 2 + n]], {n, 2, 10}] (* Vladimir Reshetnikov, Oct 17 2015 *)
Table[2*Sum[(2^k - 1) * Abs[StirlingS1[n, k]] * BernoulliB[k], {k, 0, n}], {n, 3,
25}] (* Vaclav Kotesovec, Oct 04 2022 *)

PARI

a(n)=sum(i=1,n-1,i!*(n-i)!) \\ Jon Perry, May 06 2006

PARI

{ a=0; for (n = 2, 200, write("b059371.txt", n, " ", a = (n - 1)! + a*(n + 1)/2); ) } \\ Harry J. Smith, Jun 26 2009

Formula

E.g.f.: (x^2-2*x-2*log(1-x))/(x-2)^2. - Vladeta Jovovic, May 04 2003

a(n) = Sum_{i=1..n-1} i!*(n-i)!. E.g., a(6) = 1!*5!+2!4!+3!3!+4!2!+5!1! = 120+48+36+48+120 = 372. - Jon Perry, May 06 2006

a(n) = 2*Integral_{t>=0}t^n*exp(-t)*(t*exp(-t)*Ei(t)-1), with Ei the exponential integral function.

Recurrence: 2*a(n) = (3*n-1)*a(n-1) - (n-1)*n*a(n-2). - Vaclav Kotesovec, Aug 11 2013

a(n) ~ 2*(n-1)!. - Vaclav Kotesovec, Aug 11 2013

a(n) = -2*n! - 2*(n+1)!*Re(LerchPhi(2, 1, 2 + n)). - Vladimir Reshetnikov, Oct 17 2015

a(n) = n!*Re(hypergeom([1,1],[n+2],2) - 1). - Vladimir Reshetnikov, Oct 19 2015

Extensions

Better description from Vladeta Jovovic, May 04 2003

More terms from Larry Reeves (larryr(AT)acm.org), Jan 31 2001