A193445 a(n) = n! * Sum_{d|n} H(d)*H(n/d), where H(n) is the n-th harmonic number.

1, 6, 22, 154, 548, 7488, 26136, 471168, 3272832, 46114560, 241087680, 10152587520, 39605518080, 1245053859840, 19626466406400, 402874746624000, 2446811181158400, 156604969130803200, 863130293635276800, 62029933697765376000, 858218507492806656000

Offset: 1

Author

Paul D. Hanna, Jul 25 2011

Keywords

• nonn

Examples

			E.g.f.: A(x) = x + 6*x^2/2! + 22*x^3/3! + 154*x^4/4! + 548*x^5/5! + 7488*x^6/6! + ... where A(x) = -H(1)*log(1-x)/(1-x) - H(2)*log(1-x^2)/(1-x^2) - H(3)*log(1-x^3)/(1-x^3) + ...
More explicitly,
A(x) = -(1)*log(1-x)/(1-x) - (1+1/2)*log(1-x^2)/(1-x^2) - (1+1/2+1/3)*log(1-x^3)/(1-x^3) - (1+1/2+1/3+1/4)*log(1-x^4)/(1-x^4) + ...
Illustration of terms:
a(2) = 2!*(1*(1+1/2) + (1+1/2)*1) = 6;
a(3) = 3!*(1*(1+1/2+1/3) + (1+1/2+1/3)*1) = 22;
a(4) = 4!*(1*(1+1/2+1/3+1/4) + (1+1/2)*(1+1/2) + (1+1/2+1/3+1/4)*1) = 154;
a(6) = 6!*(1*(1+1/2+1/3+1/4+1/5+1/6) + (1+1/2)*(1+1/2+1/3) + (1+1/2+1/3)*(1+1/2) + (1+1/2+1/3+1/4+1/5+1/6)*1) = 7488; ...
		

Crossrefs

Cf. A001008, A002805, A067710, A193446.

Programs

Mathematica

a[n_] := n! * DivisorSum[n, HarmonicNumber[#] * HarmonicNumber[n/#] &]; Array[a, 20] (* Amiram Eldar, Aug 18 2023 *)

PARI

{a(n)=if(n<1,0,n!*sumdiv(n,d,sum(j=1,d,1/j)*sum(k=1,n/d,1/k)))}

PARI

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

Formula

E.g.f.: Sum_{n>=1} -H(n) * log(1 - x^n) / (1 - x^n) = Sum_{n>=1} a(n)*x^n/n!, where H(n) is the n-th harmonic number.

a(n) = n! * Sum_{d|n} (Sum_{j=1..d} 1/j)*(Sum_{k=1..n/d} 1/k).

Extensions

a(21) from Amiram Eldar, Aug 18 2023