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

0,2

Links

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

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

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 *)

Prog

(PARI) a(n)=polcoeff((1-(1-16*x+x^2*O(x^n))^(1/8))/(2*x), n, x)
(Magma) R<x>:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (1-(1-16*x)^(1/8))/(2*x) )); // G. C. Greubel, Sep 17 2019
(Sage)
def A097184_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P((1-(1-16*x)^(1/8))/(2*x)).list()
A097184_list(20) # G. C. Greubel, Sep 17 2019

Crossrefs

Cf. A097181, A097182, A097183.

Sequence in context: A078246 A268301 A228927 * A002296 A027394 A346767

Adjacent sequences: A097181 A097182 A097183 * A097185 A097186 A097187

Keyword

nonn

Author

Paul D. Hanna, Aug 03 2004

Extensions

More terms from Vincenzo Librandi, Feb 10 2014

Status

approved