A097192 Main diagonal of triangle A097190, in which the n-th row polynomial R_n(y) is formed from the initial (n+1) terms of g.f. A097191(y)^(n+1), where R_n(1/3) = 9^n for all n>=0.

1, 24, 612, 15912, 417690, 11027016, 292215924, 7764594552, 206732329947, 5512862131920, 147193418922264, 3934078651195056, 105236603919467748, 2817102935690367408, 75458114348849127000, 2022277464549156603600

Offset: 0

Author

Paul D. Hanna, Aug 03 2004

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..695

Crossrefs

Cf. A097190, A097191, A097193.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( 1/(1-27*x)^(8/9) )); // G. C. Greubel, Sep 17 2019

Maple

seq(coeff(series(1/(1-27*x)^(8/9), x, n+1), x, n), n = 0 ..20); # G. C. Greubel, Sep 17 2019

Mathematica

CoefficientList[Series[(1-27*x)^(-8/9), {x,0,20}], x] (* G. C. Greubel, Sep 17 2019 *)

PARI

a(n)=polcoeff(1/(1-27*x+x*O(x^n))^(8/9),n,x)

Sage

def A097192_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P(1/(1-27*x)^(8/9)).list()
A097192_list(20) # G. C. Greubel, Sep 17 2019

Formula

G.f.: A(x) = 1/(1-27*x)^(8/9).

a(n) = (n+1)*A097193(n).

Conjecture: n*a(n) +3*(1-9*n)*a(n-1) = 0. - R. J. Mathar, Nov 16 2012