A097184 G.f. A(x) satisfies A097182(x*A(x)) = A(x) and so equals the ratio of the g.f.s of any two adjacent diagonals of triangle A097181.

1, 7, 70, 805, 9982, 129766, 1742572, 23960365, 335445110, 4763320562, 68418604436, 992069764322, 14499481170860, 213349508656940, 3157572728122712, 46968894330825341, 701770538825272742, 10526558082379091130, 158452400608443161220

Offset: 0

Author

Paul D. Hanna, Aug 03 2004

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A097181, A097182, A097183.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (1-(1-16*x)^(1/8))/(2*x) )); // G. C. Greubel, Sep 17 2019

Maple

seq(coeff(series((1-(1-16*x)^(1/8))/(2*x), x, n+2), x, n), n = 0..20); # G. C. Greubel, Sep 17 2019

Mathematica

CoefficientList[Series[(1-(1-16*x)^(1/8))/(2*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 09 2014 *)
Table[FullSimplify[16^n*Gamma[n+7/8]/(Gamma[7/8]*Gamma[n+2])], {n, 0, 20}] (* Vaclav Kotesovec, Feb 09 2014 *)

PARI

a(n)=polcoeff((1-(1-16*x+x^2*O(x^n))^(1/8))/(2*x),n,x)

Sage

def A097184_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P((1-(1-16*x)^(1/8))/(2*x)).list()
A097184_list(20) # G. C. Greubel, Sep 17 2019

Formula

G.f.: A(x) = (1-(1-16*x)^(1/8))/(2*x).

G.f.: A(x) = (1/x)*(series reversion of x/A097182(x)).

a(n) = A097183(n)/(n+1).

D-finite with recurrence: (n+1)*a(n) +2*(-8*n+1)*a(n-1)=0. - R. J. Mathar, Nov 16 2012

a(n) = 16^n * Gamma(n+7/8) / (Gamma(7/8) * Gamma(n+2)). - Vaclav Kotesovec, Feb 09 2014

a(n) ~ 16^n / (Gamma(7/8) * n^(9/8)). - Vaclav Kotesovec, Feb 09 2014

Extensions

More terms from Vincenzo Librandi, Feb 10 2014