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

1,2

Links

Table of n, a(n) for n=1..21.

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).

Example

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; ...

Mathematica

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

Prog

(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))}

Crossrefs

Cf. A001008, A002805, A067710, A193446.

Sequence in context: A213130 A002137 A009361 * A075759 A000993 A028406

Adjacent sequences: A193442 A193443 A193444 * A193446 A193447 A193448

Keyword

nonn

Author

Paul D. Hanna, Jul 25 2011

Extensions

a(21) from Amiram Eldar, Aug 18 2023

Status

approved