A132374 - OEIS

A132374

Expansion of c(7*x^2)/(1 - x*c(7*x^2)), where c(x) is the g.f. of A000108.

1, 1, 8, 15, 120, 274, 2192, 5531, 44248, 118686, 949488, 2654646, 21237168, 61189668, 489517344, 1443039123, 11544312984, 34648845862, 277190766896, 844131474530, 6753051796240, 20813234394492, 166505875155936, 518373091849502

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OFFSET

0,3

COMMENTS

Hankel transform is 7^C(n+1,2).

Series reversion of x*(1+x)*(1+2*x+8*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) * 7^(n-k).

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

a(n) = 4*( 2*(n+1)*a(n-1) + 7*(n-2)*a(n-2) - 56*(n-2)*a(n-3) )/(n+1).

G.f.: (1 - sqrt(1 - 28*x^2))/(14*x^2 - x*(1 - sqrt(1 - 28*x^2))). (End)

MATHEMATICA

CoefficientList[Series[(1-Sqrt[1-28*x^2])/(14*x^2 -x*(1-Sqrt[1-28*x^2])), {x, 0, 40}], x] (* G. C. Greubel, Nov 08 2022 *)

PROG

(Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(1-28*x^2))/(14*x^2 -x*(1-Sqrt(1-28*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 A132374(n): return sum(7^(n-k)*A120730(n, k) for k in range(n+1))

[A132374(n) for n in range(51)] # G. C. Greubel, Nov 08 2022

CROSSREFS

Cf. A000108, A001405, A120730, A126087, A128386, A121724, A128387, A121725.

Sequence in context: A189003 A110294 A110459 * A346314 A361710 A234534

Adjacent sequences: A132371 A132372 A132373 * A132375 A132376 A132377

KEYWORD

easy,nonn

AUTHOR

Philippe Deléham, Nov 10 2007

STATUS

approved