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
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Oct 16 2018
STATUS
approved