A097194 Row sums 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, 25, 649, 17065, 451621, 11998801, 319623445, 8530126057, 227974775239, 6099550226965, 163340461497907, 4377292845062689, 117376545230379631, 3149059523347103293, 84522568856319875179, 2269506752111508954553

Offset: 0

Author

Paul D. Hanna, Aug 03 2004

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..690

Crossrefs

Cf. A097190.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( 3/((1-27*x) +2*(1-27*x)^(8/9)) )); // G. C. Greubel, Sep 17 2019

Maple

seq(coeff(series(3/((1-27*x) +2*(1-27*x)^(8/9)), x, n+1), x, n), n = 0 ..20); # G. C. Greubel, Sep 17 2019

Mathematica

CoefficientList[Series[3/((1-27*x) +2*(1-27*x)^(8/9)), {x,0,20}], x] (* G. C. Greubel, Sep 17 2019 *)

PARI

a(n)=polcoeff(3/((1-27*x) + 2*(1-27*x+x*O(x^n))^(8/9)),n,x)

Sage

def A097194_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P(3/((1-27*x) +2*(1-27*x)^(8/9))).list()
A097194_list(20) # G. C. Greubel, Sep 17 2019

Formula

G.f.: A(x) = 3/((1-27*x) + 2*(1-27*x)^(8/9)).

G.f.: A(x, y) = A097192(x)/(1 - x*A097193(x)).