A166587 A signed variant of the Motzkin numbers.

1, 1, -1, 2, -4, 9, -21, 51, -127, 323, -835, 2188, -5798, 15511, -41835, 113634, -310572, 853467, -2356779, 6536382, -18199284, 50852019, -142547559, 400763223, -1129760415, 3192727797, -9043402501, 25669818476, -73007772802

Offset

0,4

Comments

Hankel transform is A131713. Binomial transform is A166588.

[a(n+1)] = [1,-1,2,-4,9,...] is the inverse binomial transform of A126120. - Philippe Deléham, Nov 29 2009

Links

G. C. Greubel, Table of n, a(n) for n = 0..500

Formula

G.f.: (1+3x-sqrt(1+2x-3x^2))/(2x); (1+3x)/(1+2x-x^2/(1+x-x^2/(1+x-x^2/(1+x-x^2/(1+...))))) (continued fraction).

a(n) = 0^n + Sum_{k=0..n} binomial(n-1, k-1)*(-3)^(n-k)*A000108(k).

G.f.: (1+3*x-sqrt(1+2*x-3*x^2))/(2x) = (3-1/G(0))/2 ; G(k) = 1+2*x/(1-x/(1-x/(1+2*x/(1+x/(2+x/G(k+1)))))) ; (continued fraction). - Sergei N. Gladkovskii, Dec 11 2011

Conjecture: n*(n+1)*a(n) + n*(n+1)*a(n-1) - (5*n-3)*(n-2)*a(n-2) + 3*(n-2)*(n-3)*a(n-3) = 0. - R. J. Mathar, Nov 15 2012

G.f. G(x) satisfies (3 x^2 - 2 x^2 - x) G'(x) - (x+1) G(x) + 3 x + 1 = 0, from which follows 3*n*a(n) + (-3-2*n)*a(1+n) + (-3-n)*a(n+2) = 0 as well as Mathar's conjecture. - Robert Israel, May 17 2016

E.g.f.: 1 + Integral (exp(-x) * BesselI(1,2*x) / x) dx. - Ilya Gutkovskiy, Jun 01 2020

Example

G.f. = 1 + x - x^2 + 2*x^3 - 4*x^4 + 9*x^5 - 21*x^6 + 51*x^7 - 127*x^8 + ...

Maple

f:= gfun:-rectoproc({3*n*a(n)+(-3-2*n)*a(1+n)+(-3-n)*a(n+2)=0, a(0) = 1, a(1) = 1}, a(n), remember):
map(f, [$0..100]); # Robert Israel, May 17 2016
with(PolynomialTools): CoefficientList(convert(taylor((1 + 3*x - sqrt(1 + 2*x - 3*x^2))/2/x, x = 0, 33), polynom), x); # Taras Goy, Aug 07 2017

Mathematica

CoefficientList[Series[(1 + 3*t - Sqrt[1 + 2*t - 3*t^2])/(2 t), {t, 0, 50}], t] (* G. C. Greubel, May 17 2016 *)

Crossrefs

Sequence in context: A094287 A094288 A168051 * A292440 A168049 A001006

Adjacent sequences: A166584 A166585 A166586 * A166588 A166589 A166590

Keyword

easy,sign

Author

Paul Barry, Oct 17 2009

Status

approved