A320589 Expansion of (1/(1 + x)) * Sum_{k>=1} k*x^k/(x^k + (1 + x)^k).

1, -1, 4, -13, 36, -88, 197, -421, 895, -1946, 4346, -9832, 22140, -49043, 106389, -226213, 473366, -980413, 2022418, -4179198, 8687753, -18201140, 38398455, -81343408, 172383461, -364158198, 764854519, -1595107695, 3302884966, -6796646603, 13921482698, -28437025029, 58034908034

Offset

1,3

Comments

Inverse binomial transform of A000593.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Vaclav Kotesovec, Plot of |a(n)|/(n*2^n) for n = 1..10000

N. J. A. Sloane, Transforms

Formula

G.f.: (theta_3(x/(1 + x))^4 + theta_2(x/(1 + x))^4 - 1)/(24*(1 + x)), where theta_() is the Jacobi theta function.

L.g.f.: Sum_{k>=1} A000593(k)*x^k/(k*(1 + x)^k) = Sum_{n>=1} a(n)*x^n/n.

a(n) = Sum_{k=1..n} (-1)^(n-k)*binomial(n,k)*A000593(k).

Conjecture: a(n) ~ -(-1)^n * c * 2^n * n, where c = Pi^2/48 = 0.205616758356... - Vaclav Kotesovec, Jun 26 2019

Maple

seq(coeff(series((1/(1+x))*add(k*x^k/(x^k+(1+x)^k), k=1..n), x, n+1), x, n), n = 1 .. 35); # Muniru A Asiru, Oct 16 2018

Mathematica

nmax = 33; Rest[CoefficientList[Series[1/(1 + x) Sum[k x^k/(x^k + (1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 33; Rest[CoefficientList[Series[(EllipticTheta[3, 0, x/(1 + x)]^4 + EllipticTheta[2, 0, x/(1 + x)]^4 - 1)/(24 (1 + x)), {x, 0, nmax}], x]]
Table[Sum[(-1)^(n - k) Binomial[n, k] Sum[(-1)^(k/d + 1) d, {d, Divisors[k]}], {k, n}], {n, 33}]

Prog

(PARI) m=50; x='x+O('x^m); Vec((1/(1 + x))*sum(k=1, m+2, k*x^k/(x^k + (1 + x)^k))) \\ G. C. Greubel, Oct 29 2018
(Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1/(1 + x))*(&+[k*x^k/(x^k + (1 + x)^k): k in [1..(m+2)]]) )); // G. C. Greubel, Oct 29 2018

Crossrefs

Cf. A000593, A320568, A320586, A320591.

Sequence in context: A272556 A173723 A002727 * A328542 A036629 A350642

Adjacent sequences: A320586 A320587 A320588 * A320590 A320591 A320592

Keyword

sign

Author

Ilya Gutkovskiy, Oct 16 2018

Status

approved