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A222115
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a(n) = 1 + Sum_{k=1..n} binomial(n,k) * sigma(k).
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3
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2, 6, 17, 46, 117, 285, 674, 1558, 3536, 7911, 17503, 38377, 83501, 180480, 387882, 829606, 1766999, 3749766, 7931115, 16724871, 35173778, 73794661, 154485528, 322771345, 673155142, 1401536935, 2913490376, 6047714600, 12536770559, 25956242580, 53678385267, 110889844998
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OFFSET
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1,1
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COMMENTS
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Here sigma(n) is the sum of divisors of n (A000203).
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LINKS
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FORMULA
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Logarithmic derivative of the binomial transform of the partition numbers (A218481).
L.g.f.: -log(1-x) + Sum_{n>=1} sigma(n) * x^n/(1-x)^n / n.
L.g.f.: -log(1-x) + Sum_{n>=1} x^n/((1-x)^n - x^n) / n.
L.g.f.: -log(1-x) + Sum_{n>=1} n*log(1-x) - log((1-x)^n - x^n).
L.g.f.: -log(1-x) + Sum_{n>=1} A001511(n) * log(1 + x^n/(1-x)^n), where 2^A001511(n) is the highest power of 2 that divides 2*n.
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EXAMPLE
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L.g.f.: L(x) = 2*x + 6*x^2/2 + 17*x^3/3 + 46*x^4/4 + 117*x^5/5 + 285*x^6/6 +...
where
exp(L(x)) = 1 + 2*x + 5*x^2 + 13*x^3 + 34*x^4 + 88*x^5 + 225*x^6 + 569*x^7 +...+ A218481(n)*x^n +...
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MATHEMATICA
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Table[Sum[Binomial[n, k]DivisorSigma[1, k], {k, n}], {n, 40}]+1 (* Harvey P. Dale, Jul 21 2015 *)
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PROG
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(PARI) {a(n)=1+sum(k=1, n, binomial(n, k)*sigma(k))}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n)=local(X=x+x*O(x^n)); n*polcoeff(-log(1-X)+sum(m=1, n+1, x^m/((1-x)^m-X^m)/m), n)}
(PARI) {a(n)=local(X=x+x*O(x^n)); n*polcoeff(-log(1-X)+sum(k=1, n, k*log(1-X)-log((1-x)^k-X^k)), n)}
(PARI) {a(n)=local(X=x+x*O(x^n)); n*polcoeff(-log(1-X)+sum(m=1, n+1, sigma(m)*x^m/(1-X)^m/m), n)}
(PARI) {a(n)=local(X=x+x*O(x^n)); n*polcoeff(-log(1-X)+sum(k=1, n, valuation(2*k, 2)*log(1 + x^k/(1-X)^k)), 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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