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A206764 a(n) = Sum_{k=1..n} binomial(n,k) * sigma(n,k) * (-1)^(n-k).

%I A206764 #12 Oct 25 2024 03:17:03
%S A206764 1,-1,10,79,1026,15686,279938,5771359,134218243,3487832974,
%T A206764 100000000002,3138673052878,106993205379074,3937454749863382,
%U A206764 155568096631586820,6568441588686506943,295147905179352825858,14063102470280932000757,708235345355337676357634
%N A206764 a(n) = Sum_{k=1..n} binomial(n,k) * sigma(n,k) * (-1)^(n-k).
%C A206764 Here sigma(n,k) equals the sum of the k-th powers of the divisors of n.
%F A206764 a(n) ~ exp(-1) * n^n. - _Vaclav Kotesovec_, Oct 25 2024
%e A206764 L.g.f.: L(x) = x - x^2/2 + 10*x^3/3 + 79*x^4/4 + 1026*x^5/5 + 15686*x^6/6 +...
%e A206764 Exponentiation yields the g.f. of A206763:
%e A206764 exp(L(x)) = 1 + x + 3*x^3 + 23*x^4 + 225*x^5 + 2824*x^6 + 42670*x^7 +...
%e A206764 Illustration of terms.
%e A206764 a(2) = -2*sigma(2,1) + 1*sigma(2,2) = -2*3 + 1*5 = -1;
%e A206764 a(3) = 3*sigma(3,1) - 3*sigma(3,2) + 1*sigma(3,3) = 3*4 - 3*10 + 1*28 = 10;
%e A206764 a(4) = -4*sigma(4,1) + 6*sigma(4,2) - 4*sigma(4,3) + 1*sigma(4,4) = -4*7 + 6*21 - 4*73 + 1*273 = 79.
%t A206764 Table[Sum[Binomial[n, k] * DivisorSigma[k, n] * (-1)^(n-k), {k, 1, n}], {n, 1, 20}] (* _Vaclav Kotesovec_, Oct 25 2024 *)
%o A206764 (PARI) {a(n)=sum(k=1, n, binomial(n, k)*sigma(n, k)*(-1)^(n-k))}
%o A206764 (PARI) {a(n)=n*polcoeff(sum(k=1, n, (1/k)*log((1-(-x)^k)/(1-(k-1)^k*x^k +x*O(x^n)))), n)}
%o A206764 for(n=1,21,print1(a(n),", "))
%Y A206764 Cf. A206763 (exp), A205815, A205812.
%K A206764 sign,easy
%O A206764 1,3
%A A206764 _Paul D. Hanna_, Feb 12 2012