A190736 Diagonal sums of the Riordan matrix A121576.

1, 2, 7, 29, 139, 731, 4096, 24005, 145420, 903503, 5726290, 36878978, 240663403, 1587928511, 10575884599, 71005972250, 480071241463, 3265685620913, 22335284505529, 153496543690226, 1059443187603955, 7340794592800628, 51042913856490028

Offset: 0

Author

Emanuele Munarini, May 18 2011

Keywords

• nonn

Links

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

Crossrefs

Cf. A121576.

Programs

Magma

m:=25; R:=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

Mathematica

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

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

Formula

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)).

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)).

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).

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