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

0,3

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

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)

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 *)

Prog

(Magma) R<x>:=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
(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

Crossrefs

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

Sequence in context: A177166 A073221 A110462 * A110360 A371031 A123836

Adjacent sequences: A132372 A132373 A132374 * A132376 A132377 A132378

Keyword

easy,nonn

Author

Philippe Deléham, Nov 10 2007

Status

approved