A132373 Expansion of c(6*x^2)/(1-x*c(6*x^2)), where c(x) is the g.f. of A000108.

1, 1, 7, 13, 91, 205, 1435, 3565, 24955, 65821, 460747, 1265677, 8859739, 25066621, 175466347, 507709165, 3553964155, 10466643805, 73266506635, 218878998733, 1532152991131, 4631531585341, 32420721097387, 98980721277613, 692865048943291, 2133274258946845

Offset: 0

Author

Philippe Deléham, Nov 10 2007

Keywords

• easy
• nonn

Comments

Hankel transform is 6^C(n+1, 2).

Series reversion of (1+x)/(1 + 2*x + 7*x^2). [Corrected by R. J. Mathar, Nov 19 2009]

Links

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

Crossrefs

Cf. A000012, A000108, A001405, A120730, A121724, A126087, A128386, A128387.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(1-24*x^2))/(12*x^2-x*(1-Sqrt(1-24*x^2))) )); // G. C. Greubel, Nov 07 2022

Mathematica

CoefficientList[Series[(1-Sqrt[1-24*x^2])/(12*x^2 -x*(1-Sqrt[1-24*x^2])), {x, 0, 40}], x] (* G. C. Greubel, Nov 07 2022 *)

SageMath

def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
def A132373(n): return sum(6^(n-k)*A120730(n,k) for k in range(n+1))
[A132373(n) for n in range(51)] # G. C. Greubel, Nov 07 2022

Formula

a(n) = Sum_{k=0..n} A120730(n,k) * 6^(n-k).

From G. C. Greubel, Nov 07 2022: (Start)

G.f.: (1 - sqrt(1-24*x^2))/(12*x^2 - x*(1 - sqrt(1-24*x^2))).

a(n) = ( 7*(n+1)*a(n-1) + 24*(n-2)*a(n-2) - 168*(n-2)*a(n-3) )/(n+1). (End)

Extensions

Terms beyond a(7) added by R. J. Mathar, Nov 19 2009