A320568 a(n) = Sum_{k=1..n} (-1)^(n-k)*binomial(n,k)*sigma(k).

1, 1, -2, 5, -14, 40, -111, 293, -731, 1726, -3882, 8408, -17796, 37423, -79337, 170917, -373812, 823585, -1809844, 3934750, -8424747, 17749392, -36874749, 75862016, -155359339, 318331170, -655146929, 1356952623, -2828151136, 5920984735, -12420140296, 26036563525

Offset

1,3

Comments

Inverse binomial transform of A000203.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..3000

Vaclav Kotesovec, Graph abs(a(n)) / (n*2^n)

N. J. A. Sloane, Transforms

Formula

G.f.: (1/(1 + x))*Sum_{k>=1} k*x^k/((1 + x)^k - x^k).

L.g.f.: Sum_{k>=1} sigma(k)*x^k/(k*(1 + x)^k) = Sum_{n>=1} a(n)*x^n/n.

Conjecture: a(n) ~ (-1)^n * Pi^2/48 * n * 2^n. - Vaclav Kotesovec, Oct 16 2018

Maple

seq(add((-1)^(n-k)*binomial(n, k)*sigma(k), k=1..n), n=1..32); # Paolo P. Lava, Oct 25 2018

Mathematica

Table[Sum[(-1)^(n - k) Binomial[n, k] DivisorSigma[1, k], {k, n}], {n, 32}]
nmax = 32; Rest[CoefficientList[Series[1/(1 + x) Sum[k x^k/((1 + x)^k - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]

Prog

(GAP) Flat(List([1..10], n->Sum([1..n], k->(-1)^(n-k)*Binomial(n, k)*Sigma(k))));  # Muniru A Asiru, Oct 15 2018

Crossrefs

Cf. A000203, A038200, A185003.

Sequence in context: A148318 A148319 A126219 * A111110 A296516 A111109

Adjacent sequences: A320565 A320566 A320567 * A320569 A320570 A320571

Keyword

sign

Author

Ilya Gutkovskiy, Oct 15 2018

Status

approved