A103137 First column of inverse of Delannoy triangle.

1, -1, 2, -6, 22, -90, 394, -1806, 8558, -41586, 206098, -1037718, 5293446, -27297738, 142078746, -745387038, 3937603038, -20927156706, 111818026018, -600318853926, 3236724317174, -17518619320890, 95149655201962, -518431875418926, 2832923350929742, -15521467648875090

Offset: 0

Author

Paul Barry, Jan 24 2005

Keywords

• easy
• sign

Comments

First column of A103136. The positive sequence has g.f. 1+xS(x). It is the first column of the inverse of the signed Delannoy triangle which has general term T(n,k)=if(k<=n, sum{j=0..k, 2^j*C(n-k,j)C(k,j)}(-1)^(n-k),0).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

A minor variation of A006318.

Programs

Mathematica

CoefficientList[Series[1+x*(1+x-(1+6*x+x^2)^(1/2))/(2*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

Formula

G.f.: 1-xS(-x), where S(x) is the g.f. of the large Schroeder numbers A006318.

Conjecture: n*a(n) +3*(2*n-3)*a(n-1) +(n-3)*a(n-2)=0. - R. J. Mathar, Nov 26 2012

a(n) ~ (-1)^n * sqrt(3*sqrt(2)-4) * (3+2*sqrt(2))^n / (2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Feb 13 2014