A116394 - OEIS

%I #13 Sep 08 2022 08:45:24

%S 1,1,4,11,33,100,305,937,2890,8943,27741,86216,268355,836297,2608818,

%T 8144875,25446229,79545148,248780979,778400001,2436380402,7628211951,

%U 23890103153,74836927720,234478937321,734802907841,2303073316042

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

%H G. C. Greubel, <a href="/A116394/b116394.txt">Table of n, a(n) for n = 0..1000</a>

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

%F 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

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

%o (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

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/((1+x)*Sqrt(1-2*x-3*x^2) - x) )); // _G. C. Greubel_, May 28 2019

%o (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

%Y Diagonal sums of number triangle A116392.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Feb 12 2006