A190736 - OEIS

%I #14 Sep 08 2022 08:45:57

%S 1,2,7,29,139,731,4096,24005,145420,903503,5726290,36878978,240663403,

%T 1587928511,10575884599,71005972250,480071241463,3265685620913,

%U 22335284505529,153496543690226,1059443187603955,7340794592800628,51042913856490028

%N Diagonal sums of the Riordan matrix A121576.

%H G. C. Greubel, <a href="/A190736/b190736.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = [x^n](1-2*x-2*x^2)*(1+2*x)^(n+1)/((1+2*x-x^2+x^3)(1-x)^(n+1)).

%F G.f.: (4-5*x-2*x^2-(2+x)*sqrt(1-8*x+4*x^2))/(2*(1-x+2*x^2+x^3)).

%F Recurrence: 0 = 6*(n^2+17*n+72)*a(n+9) - (35*n^2+577*n+2376)*a(n+8) - (81*n^2+835*n+1856)*a(n+7) + (101*n^2+1017*n+2164)*a(n+6) - 2*(151*n^2+1883*n+5970)*a(n+5) - 2*(33*n^2+458*n+1528)*a(n+4) + (47*n^2+567*n+1564)*a(n+3) - 2*(7*n^2-16*n-120)*a(n+2) + 4*(3*n^2+8*n+4)*a(n+1) + 8*(n^2+3*n+2)*a(n).

%F Conjecture: n*(11*n-35)*a(n) + 3*(-33*n^2+149*n-136)*a(n-1) +2*(77*n^2-377*n+396)*a(n-2) +(-209*n^2+1061*n-1200)*a(n-3) +12*(-11*n+30)*a(n-4) +4*(11*n-24)*(n-4)*a(n-5)=0. - _R. J. Mathar_, Jul 24 2012

%t CoefficientList[Series[(4-5x-2x^2-(2+x)Sqrt[1-8x+4x^2])/(2(1-x+2x^2 +x^3) ),{x,0,22}],x]

%o (PARI) x='x+O('x^30); Vec((4-5*x-2*x^2-(2+x)*sqrt(1-8*x+4*x^2))/(2*(1-x+2*x^2+x^3))) \\ _G. C. Greubel_, Apr 23 2018

%o (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((4-5*x-2*x^2-(2+x)*Sqrt(1-8*x+4*x^2))/(2*(1-x+2*x^2+x^3)))); // _G. C. Greubel_, Apr 23 2018

%Y Cf. A121576.

%K nonn

%O 0,2

%A _Emanuele Munarini_, May 18 2011