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A193445
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a(n) = n! * Sum_{d|n} H(d)*H(n/d), where H(n) is the n-th harmonic number.
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1
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1, 6, 22, 154, 548, 7488, 26136, 471168, 3272832, 46114560, 241087680, 10152587520, 39605518080, 1245053859840, 19626466406400, 402874746624000, 2446811181158400, 156604969130803200, 863130293635276800, 62029933697765376000, 858218507492806656000
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OFFSET
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1,2
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LINKS
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FORMULA
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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).
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EXAMPLE
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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; ...
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MATHEMATICA
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a[n_] := n! * DivisorSum[n, HarmonicNumber[#] * HarmonicNumber[n/#] &]; Array[a, 20] (* Amiram Eldar, Aug 18 2023 *)
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PROG
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(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))}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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