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

%I A193445 #13 Aug 20 2023 10:51:00
%S A193445 1,6,22,154,548,7488,26136,471168,3272832,46114560,241087680,
%T A193445 10152587520,39605518080,1245053859840,19626466406400,402874746624000,
%U A193445 2446811181158400,156604969130803200,863130293635276800,62029933697765376000,858218507492806656000
%N A193445 a(n) = n! * Sum_{d|n} H(d)*H(n/d), where H(n) is the n-th harmonic number.
%F A193445 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.
%F A193445 a(n) = n! * Sum_{d|n} (Sum_{j=1..d} 1/j)*(Sum_{k=1..n/d} 1/k).
%e A193445 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) + ...
%e A193445 More explicitly,
%e A193445 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) + ...
%e A193445 Illustration of terms:
%e A193445 a(2) = 2!*(1*(1+1/2) + (1+1/2)*1) = 6;
%e A193445 a(3) = 3!*(1*(1+1/2+1/3) + (1+1/2+1/3)*1) = 22;
%e A193445 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;
%e A193445 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; ...
%t A193445 a[n_] := n! * DivisorSum[n, HarmonicNumber[#] * HarmonicNumber[n/#] &]; Array[a, 20] (* _Amiram Eldar_, Aug 18 2023 *)
%o A193445 (PARI) {a(n)=if(n<1,0,n!*sumdiv(n,d,sum(j=1,d,1/j)*sum(k=1,n/d,1/k)))}
%o A193445 (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))}
%Y A193445 Cf. A001008, A002805, A067710, A193446.
%K A193445 nonn
%O A193445 1,2
%A A193445 _Paul D. Hanna_, Jul 25 2011
%E A193445 a(21) from _Amiram Eldar_, Aug 18 2023