A196532 a(n) = (n+1)!*(H(n)+H(n+1)), where H(n) = Sum_{k=1..n} 1/k is the n-th harmonic number.

1, 5, 20, 94, 524, 3408, 25416, 214128, 2012832, 20894400, 237458880, 2932968960, 39126516480, 560704273920, 8591147712000, 140160890419200, 2425888391270400, 44398288688947200, 856727919929548800

Offset: 0

Author

Gary Detlefs, Oct 03 2011

Keywords

• nonn

Comments

Denominator of a(n)/n! is listed in A096620.

a(n) - (n+1)*a(n-1) = A129326(n), n > 0. - Gary Detlefs, Oct 04 2011

Links

• Robert Israel, Table of n, a(n) for n = 0..448

Programs

Maple

H:= n-> sum(1/k,k=1..n):seq((n+1)!*(H(n+1)+H(n)), n=0..20);
# Alternative:
f:= gfun:-rectoproc({a(n+3) = (3*n+8)*a(n+2)-(3*n+7)*(n+2)*a(n+1)+(n+1)*(n+2)^2*a(n),a(0)=1,a(1)=5,a(2)=20},a(n),remember):
map(f, [$0..50]); # Robert Israel, Mar 28 2018

Mathematica

Table[(n+1)!Total[HarmonicNumber[{n,n+1}]],{n,0,20}] (* Harvey P. Dale, Jul 17 2013 *)

Formula

From Robert Israel, Mar 28 2018: (Start)

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

a(n+3) = (3*n+8)*a(n+2) - (3*n+7)*(n+2)*a(n+1) + (n+1)*(n+2)^2*a(n). (End)