A116394 Expansion of 1/((1+x)*sqrt(1-2*x-3*x^2) - x).

1, 1, 4, 11, 33, 100, 305, 937, 2890, 8943, 27741, 86216, 268355, 836297, 2608818, 8144875, 25446229, 79545148, 248780979, 778400001, 2436380402, 7628211951, 23890103153, 74836927720, 234478937321, 734802907841, 2303073316042

Offset: 0

Author

Paul Barry, Feb 12 2006

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Diagonal sums of number triangle A116392.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/((1+x)*Sqrt(1-2*x-3*x^2) - x) )); // G. C. Greubel, May 28 2019

Mathematica

CoefficientList[Series[1/((1+x)*Sqrt[1-2x-3x^2] -x), {x, 0, 30}], x] (* G. C. Greubel, May 28 2019 *)

PARI

my(x='x+O('x^30)); Vec(1/((1+x)*sqrt(1-2*x-3*x^2) - x)) \\ G. C. Greubel, May 28 2019

Sage

(1/((1+x)*sqrt(1-2*x-3*x^2) - x)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 28 2019

Formula

a(n) = Sum_{k=0..floor(n/2)} A116392(n-k,k).

D-finite with recurrence: n*a(n) +2*(-n+1)*a(n-1) +2*(-5*n+6)*a(n-2) +2*(3*n-7)*a(n-3) +2*(17*n-50)*a(n-4) +6*(5*n-17)*a(n-5) +9*(n-4)*a(n-6)=0. - R. J. Mathar, Jan 23 2020