A112478 - OEIS

A112478

Expansion of (1 + x + sqrt(1 + 6*x + x^2))/2.

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

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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

D-finite with recurrence: 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

0 = a(n)*(a(n+1) + 15*a(n+2) + 4*a(n+3)) + a(n+1)*(-3*a(n+1) + 34*a(n+3) + 15*a(n+3)) + a(n+2)*(-3*a(n+2) + a(n+3)) for all integer n > 0. - Michael Somos, Jul 07 2020

From Seiichi Manyama, Oct 08 2023: (Start)

G.f. satisfies A(x) = 1 + x + x/A(x).

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(2*k-1,k) * binomial(n+k-2,n-k)/(2*k-1). (End)

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 *)

PROG

(PARI) {a(n) = polcoeff((1 + x + sqrt(1 + 6*x + x^2 + x*O(x^n)))/2, n)}; /* Michael Somos, Jul 07 2020 */

CROSSREFS

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

Row sums of number triangle A112477.