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

1, -1, 10, 79, 1026, 15686, 279938, 5771359, 134218243, 3487832974, 100000000002, 3138673052878, 106993205379074, 3937454749863382, 155568096631586820, 6568441588686506943, 295147905179352825858, 14063102470280932000757, 708235345355337676357634

Offset: 1

Author

Paul D. Hanna, Feb 12 2012

Keywords

• sign
• easy

Comments

Here sigma(n,k) equals the sum of the k-th powers of the divisors of n.

Examples

			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 +...
Exponentiation yields the g.f. of A206763:
exp(L(x)) = 1 + x + 3*x^3 + 23*x^4 + 225*x^5 + 2824*x^6 + 42670*x^7 +...
Illustration of terms.
a(2) = -2*sigma(2,1) + 1*sigma(2,2) = -2*3 + 1*5 = -1;
a(3) = 3*sigma(3,1) - 3*sigma(3,2) + 1*sigma(3,3) = 3*4 - 3*10 + 1*28 = 10;
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.
		

Crossrefs

Cf. A206763 (exp), A205815, A205812.

Programs

Mathematica

Table[Sum[Binomial[n, k] * DivisorSigma[k, n] * (-1)^(n-k), {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Oct 25 2024 *)

PARI

{a(n)=sum(k=1, n, binomial(n, k)*sigma(n, k)*(-1)^(n-k))}

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)}
for(n=1,21,print1(a(n),", "))

Formula

a(n) ~ exp(-1) * n^n. - Vaclav Kotesovec, Oct 25 2024