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

0,3

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

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)

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 *)

Prog

(Magma) R<x>:=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
(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

Crossrefs

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

Sequence in context: A110455 A177952 A320461 * A110293 A253333 A039687

Adjacent sequences: A132370 A132371 A132372 * A132374 A132375 A132376

Keyword

easy,nonn

Author

Philippe Deléham, Nov 10 2007

Extensions

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

Status

approved