A118348 Semi-diagonal (two rows below central terms) of pendular triangle A118345 and equal to the self-convolution cube of the central terms (A118346).

1, 3, 18, 121, 873, 6606, 51728, 415629, 3407391, 28388847, 239675406, 2045980440, 17629939980, 153142537440, 1339599358944, 11789960853293, 104327344928619, 927627432162129, 8283625668834238, 74259685465582569, 668054892245119353

Offset: 0

Author

Paul D. Hanna, Apr 26 2006

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A118345, A118346, A118347, A118349.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (Reversion( x*(1-2*x +Sqrt((1-2*x)*(1-6*x)))/(2*(1-2*x)) )/x)^3 )); // G. C. Greubel, Mar 17 2021

Mathematica

CoefficientList[(InverseSeries[Series[x*(1-2*x +Sqrt[(1-2*x)*(1-6*x)])/(2*(1-2*x)), {x, 0, 30}]]/x)^3, x] (* G. C. Greubel, Mar 17 2021 *)

PARI

{a(n) = polcoeff( (serreverse(x*(1-2*x+sqrt((1-2*x)*(1-6*x)+x*O(x^n)))/(2*(1-2*x)))/x)^3, n)}

Sage

def A118347_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (( x*(1-2*x +sqrt((1-2*x)*(1-6*x)))/(2*(1-2*x)) ).reverse()/x)^3 ).list()
A118347_list(31) # G. C. Greubel, Mar 17 2021

Formula

G.f.: ( series_inverse( x*(1-2*x +sqrt((1-2*x)*(1-6*x)))/(2*(1-2*x)) )/x )^3.