A193446 a(n) = n! * Sum_{k=1..n-1} H(k)*H(n-k) for n>=2, where H(n) is the n-th harmonic number.

2, 18, 142, 1160, 10208, 97720, 1018008, 11507472, 140581872, 1848236544, 26041957632, 391738883328, 6268657962240, 106361504663040, 1907768564282880, 36075270482058240, 717379367864693760, 14967427450111488000, 326956068509511168000, 7463341244426167296000

Offset: 2

Author

Paul D. Hanna, Jul 26 2011

Keywords

• nonn

Comments

A self-convolution of the harmonic numbers.

Examples

			E.g.f.: log(1-x)^2/(1-x)^2 = 2*x^2/2! + 18*x^3/3! + 142*x^4/4! + 1160*x^5/5! + ...
Illustration of initial terms:
a(2) = 2!*(1*1) = 2;
a(3) = 3!*(1*(1+1/2) + (1+1/2)*1) = 18;
a(4) = 4!*(1*(1+1/2+1/3) + (1+1/2)*(1+1/2) + (1+1/2+1/3)*1) = 142;
a(5) = 5!*(1*(1+1/2+1/3+1/4) + (1+1/2)*(1+1/2+1/3) + (1+1/2+1/3)*(1+1/2) + (1+1/2+1/3+1/4)*1) = 1160; ...
		

Links

• Eric Weisstein's World of Mathematics, Harmonic Number.

Crossrefs

Cf. A001008, A002805, A001706, A193445.

Programs

Mathematica

a[n_] := n! * Sum[HarmonicNumber[k] * HarmonicNumber[n-k], {k, 1, n-1}]; Array[a, 20, 2] (* Amiram Eldar, Aug 18 2023 *)

PARI

{a(n)=if(n<2,0,n!*sum(k=1,n,sum(i=1,k,1/i)*sum(j=1,n-k,1/j)))}

PARI

{a(n)=if(n<0,0,n!*polcoeff(log(1-x+x*O(x^n))^2/(1-x+x*O(x^n))^2,n))}

Formula

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

a(n) = n! * Sum_{k=1..n-1} (Sum_{i=1..k} 1/i)*(Sum_{j=1..n-k} 1/j) for n>=2.

Extensions

a(21) from Amiram Eldar, Aug 18 2023