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A167138
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G.f.: Sum_{n>=0} A167137(n)^2 * log(1+x)^n/n! where Sum_{n>=0} A167137(n)*log(1+x)^n/n! = g.f. of the partition numbers (A000041).
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4
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1, 1, 12, 148, 2523, 48996, 1127354, 29348080, 849632392, 27096593838, 943340417806, 35501579861404, 1434531966551084, 61939404662074706, 2844544965703554566, 138338597978951126666, 7098617731036257970895
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OFFSET
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0,3
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COMMENTS
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Conjecture: For all integers m > 0, Sum_{n>=0} L(n)^m * log(1+x)^n/n! is an integer series whenever Sum_{n>=0} L(n)*log(1+x)^n/n! is an integer series.
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LINKS
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FORMULA
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EXAMPLE
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G.f.: A(x) = 1 + x + 12*x^2 + 148*x^3 + 2523*x^4 + ...
Illustrate A(x) = Sum_{n>=0} A167137(n)^2*log(1+x)^n/n!:
A(x) = 1 + log(1+x) + 5^2*log(1+x)^2/2! + 31^2*log(1+x)^3/3! + 257^2*log(1+x)^4/4! + ...
where P(x), the partition function of A000041, is generated by:
P(x) = 1 + log(1+x) + 5*log(1+x)^2/2! + 31*log(1+x)^3/3! + 257*log(1+x)^4/4! + ...
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PROG
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(PARI) {Stirling2(n, k)=if(k<0|k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}
{A167137(n)=sum(k=0, n, numbpart(k)*Stirling2(n, k)*k!)}
{a(n)=polcoeff(sum(m=0, n, A167137(m)^2*log(1+x+x*O(x^n))^m/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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STATUS
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approved
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