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A205812
a(n) = Sum_{k=1..n} binomial(n,k) * sigma(n,k).
7
1, 11, 70, 719, 7806, 122534, 2097278, 43444159, 1000262653, 25997950846, 743008372734, 23312187863054, 793714773262334, 29197324076701078, 1152921975865606140, 48663045048486723199, 2185911559738696663038, 104128351926393946602653, 5242880000000000000524286
OFFSET
1,2
COMMENTS
Here sigma(n,k) equals the sum of the k-th powers of the divisors of n.
FORMULA
Logarithmic derivative of A205811.
a(n) = Sum_{d|n} ((d+1)^n - 1).
a(n) = A163190(n) - tau(n).
a(n) ~ exp(1) * n^n. - Vaclav Kotesovec, Oct 08 2016
EXAMPLE
L.g.f.: L(x) = x + 11*x^2/2 + 70*x^3/3 + 719*x^4/4 + 7806*x^5/5 +...
Exponentiation yields the g.f. of A205811:
exp(L(x)) = 1 + x + 6*x^2 + 29*x^3 + 221*x^4 + 1897*x^5 + 23502*x^6 +...
Illustration of terms.
a(2) = 2*sigma(2,1) + 1*sigma(2,2) = 2*3 + 1*5 = 11;
a(3) = 3*sigma(3,1) + 3*sigma(3,2) + 1*sigma(3,3) = 3*4 + 3*10 + 1*28 = 70;
a(4) = 4*sigma(4,1) + 6*sigma(4,2) + 4*sigma(4,3) + 1*sigma(4,3) = 4*7 + 6*21 + 4*73 + 1*273 = 719.
MATHEMATICA
Table[Sum[Binomial[n, k]*DivisorSigma[k, n], {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Oct 08 2016 *)
PROG
(PARI) {a(n)=sum(k=1, n, binomial(n, k)*sigma(n, k))}
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Paul D. Hanna, Feb 01 2012
STATUS
approved