A132375 Expansion of c(8*x^2)/(1 - x*c(8*x^2)), where c(x) is the g.f. of A000108.

1, 1, 9, 17, 153, 353, 3177, 8113, 73017, 198401, 1785609, 5060433, 45543897, 133071009, 1197639081, 3581326065, 32231934585, 98156060225, 883404542025, 2730108129937, 24570973169433, 76862217117665, 691759954058985

Offset: 0

Author

Philippe Deléham, Nov 10 2007

Keywords

• easy
• nonn

Comments

Hankel transform is 8^C(n+1, 2).

Series reversion of x*(1+x)/(1+2*x+9*x^2).

Links

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

Crossrefs

Cf. A001405, A126087, A128386, A121724, A128387, A121725.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(1-32*x^2))/(16*x^2 -x*(1-Sqrt(1-32*x^2))) )); // G. C. Greubel, Nov 08 2022

Mathematica

CoefficientList[Series[(1-Sqrt[1-32*x^2])/(16*x^2-x*(1-Sqrt[1-32*x^2])), {x,0, 40}], x] (* G. C. Greubel, Nov 08 2022 *)

SageMath

def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
def A132375(n): return sum(8^(n-k)*A120730(n,k) for k in range(n+1))
[A132375(n) for n in range(51)] # G. C. Greubel, Nov 08 2022

Formula

a(n) = Sum_{k=0..n} A120730(n,k) * 8^(n-k).

From G. C. Greubel, Nov 08 2022: (Start)

a(n) = (9*(n+1)*a(n-1) + 32*(n-2)*a(n-2) - 288*(n-2)*a(n-3))/(n+1).

G.f.: (1 - sqrt(1-32*x^2))/(16*x^2 - x*(1 - sqrt(1-32*x^2))). (End)