A060237 a(n) = n!^2 * Sum_{m=1..n}( Sum_{k=1..m} 1/(k*m) ).

1, 7, 85, 1660, 48076, 1942416, 104587344, 7245893376, 628308907776, 66687811660800, 8506654697548800, 1284292319599411200, 226530955276874956800, 46165213716463676620800

Offset: 1

Author

Leroy Quet, Mar 21 2001

Keywords

• easy
• nonn

Examples

			a(2) = 2!^2 *(1/(1*1) + 1/(1*2) + 1/(2*2)) = 7.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..252

Crossrefs

Essentially the same as A000424.

Programs

Magma

[Factorial(n)^2*(&+[(-1)^(k+1)*Binomial(n,k)/k^2: k in [1..n]]): n in [1..30]]; // G. C. Greubel, Aug 30 2018

Mathematica

Table[n!^2*Sum[(-1)^(k+1)*Binomial[n,k]/k^2, {k,1,n}], {n,1,30}] (* or *) Table[n!^2*Sum[Sum[1/(k*m), {k,1,m}], {m,1,n}], {n,1,30}](* G. C. Greubel, Aug 30 2018 *)

PARI

for(n=1,30, print1(n!^2*sum(k=1,n, (-1)^(k+1)*binomial(n,k)/k^2), ", ")) \\ G. C. Greubel, Aug 30 2018

Formula

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

From Vladeta Jovovic, Jan 29 2005: (Start)

Sum_{n>=0} a(n)*x^n/n!^2 = -dilog(1/(1-x))/(1-x).

a(n) = n!^2*Sum_{k=1..n} (-1)^(k+1)*binomial(n, k)/k^2. (End)

From Vaclav Kotesovec, Oct 23 2017: (Start)

a(n) = (3*n^2 - 3*n + 1)*a(n-1) - 3*(n-1)^4*a(n-2) + (n-2)^3*(n-1)^3*a(n-3).

a(n) ~ n!^2 * log(n)^2/2 * (1 + 2*gamma/log(n) + (Pi^2/6 + gamma^2)/log(n)^2), where gamma is the Euler-Mascheroni constant (A001620). (End)