login
Semi-diagonal (two rows below central terms) of pendular triangle A118345 and equal to the self-convolution cube of the central terms (A118346).
4

%I #7 Mar 17 2021 22:51:20

%S 1,3,18,121,873,6606,51728,415629,3407391,28388847,239675406,

%T 2045980440,17629939980,153142537440,1339599358944,11789960853293,

%U 104327344928619,927627432162129,8283625668834238,74259685465582569,668054892245119353

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

%H G. C. Greubel, <a href="/A118348/b118348.txt">Table of n, a(n) for n = 0..500</a>

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

%t 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 *)

%o (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)}

%o (Sage)

%o def A118347_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( (( x*(1-2*x +sqrt((1-2*x)*(1-6*x)))/(2*(1-2*x)) ).reverse()/x)^3 ).list()

%o A118347_list(31) # _G. C. Greubel_, Mar 17 2021

%o (Magma)

%o R<x>:=PowerSeriesRing(Rationals(), 30);

%o 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

%Y Cf. A118345, A118346, A118347, A118349.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Apr 26 2006