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A112478 Expansion of (1+x+sqrt(1+6x+x^2))/2. 4
1, 2, -2, 6, -22, 90, -394, 1806, -8558, 41586, -206098, 1037718, -5293446, 27297738, -142078746, 745387038, -3937603038, 20927156706, -111818026018, 600318853926, -3236724317174, 17518619320890, -95149655201962, 518431875418926, -2832923350929742, 15521467648875090 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This is the A-sequence for the Delannoy triangle A008288. See the W. Lang link under A006232 for Sheffer a- and z-sequences where also Riordan A- and Z-sequences are explained. O.g.f. A(y)=y/Finv(y) = 2*y/(-(1+y)+sqrt(y^2+6*y+1)) = ((1+y)+sqrt(1+6*y+y^2))/2 with Finv the inverse function of F(x)=x*(1+x)/(1-x). The o.g.f. of the Z-sequence is 1.

LINKS

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

FORMULA

G.f.: (1 +x +sqrt(1+6*x+x^2))/2. - Sergei N. Gladkovskii, Jan 04 2012

G.F.: G(0) where G(k)= 1 + x + x/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jan 04 2012

Conjecture: n*a(n) +3*(2*n-3)*a(n-1) +(n-3)*a(n-2)=0. - R. J. Mathar, Nov 24 2012

a(n) ~ (-1)^(n+1) * sqrt(3*sqrt(2)-4) * (3+2*sqrt(2))^n / (2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 12 2014

EXAMPLE

G.f. = 1 + 2*x - 2*x^2 + 6*x^3 - 22*x^4 + 90*x^5 - 394*x^6 + 1806*x^7 + ...

MATHEMATICA

CoefficientList[Series[(1+x+Sqrt[1+6*x+x^2])/2, {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

CROSSREFS

A minor variation of A006318. See A085403 for yet another version.

Row sums of number triangle A112477.

Sequence in context: A007985 A097090 A085403 * A184715 A292319 A290653

Adjacent sequences:  A112475 A112476 A112477 * A112479 A112480 A112481

KEYWORD

easy,sign

AUTHOR

Paul Barry, Sep 07 2005

STATUS

approved

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Last modified January 18 13:34 EST 2019. Contains 319271 sequences. (Running on oeis4.)