A097180 Row sums of triangle A097179, in which the n-th row polynomial R_n(y) is formed from the initial (n+1) terms of g.f. A077860(y)^(n+1), where R_n(1/2) = 4^n for all n>=0.

1, 7, 52, 395, 3036, 23506, 182904, 1428387, 11185900, 87789702, 690212744, 5434455182, 42841215704, 338081920260, 2670388231152, 21109070463267, 166980248599884, 1321686452484286, 10467203182893800, 82936871755938970

Offset: 0

Author

Paul D. Hanna, Aug 02 2004

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A077860, A097179.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( 2/((1-8*x) + (1-8*x)^(3/4)) )); // G. C. Greubel, Sep 17 2019

Maple

seq(coeff(series(2/((1-8*x) + (1-8*x)^(3/4)), x, n+1), x, n), n = 0 ..20); # G. C. Greubel, Sep 17 2019

Mathematica

CoefficientList[Series[2/((1-8*x) + (1-8*x)^(3/4)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 04 2014 *)

PARI

a(n)=polcoeff(2/((1-8*x)+(1-8*x+x*O(x^n))^(3/4)),n,x)

Sage

def A097180_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P(2/((1-8*x) + (1-8*x)^(3/4))).list()
A097180_list(20) # G. C. Greubel, Sep 17 2019

Formula

G.f.: A(x) = 2/((1-8*x) + (1-8*x)^(3/4)).

Conjecture: n*(n-1)*(n+1)*a(n) -12*n*(2*n-1)*(n-1)*a(n-1) +12*(n-1) * (16*n^2-32*n+17)*a(n-2) -16*(4*n-5)*(4*n-7)*(2*n-3)*a(n-3) = 0. - R. J. Mathar, Nov 16 2012

a(n) ~ 2^(3*n+1) / (Gamma(3/4)*n^(1/4)) * (1 - Gamma(3/4) / (n^(1/4) * sqrt(Pi))). - Vaclav Kotesovec, Feb 04 2014