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Expansion of c(6*x^2)/(1-x*c(6*x^2)), where c(x) is the g.f. of A000108.
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%I #11 Nov 09 2022 00:06:59

%S 1,1,7,13,91,205,1435,3565,24955,65821,460747,1265677,8859739,

%T 25066621,175466347,507709165,3553964155,10466643805,73266506635,

%U 218878998733,1532152991131,4631531585341,32420721097387,98980721277613,692865048943291,2133274258946845

%N Expansion of c(6*x^2)/(1-x*c(6*x^2)), where c(x) is the g.f. of A000108.

%C Hankel transform is 6^C(n+1, 2).

%C Series reversion of (1+x)/(1 + 2*x + 7*x^2). [Corrected by _R. J. Mathar_, Nov 19 2009]

%H G. C. Greubel, <a href="/A132373/b132373.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..n} A120730(n,k) * 6^(n-k).

%F From _G. C. Greubel_, Nov 07 2022: (Start)

%F G.f.: (1 - sqrt(1-24*x^2))/(12*x^2 - x*(1 - sqrt(1-24*x^2))).

%F a(n) = ( 7*(n+1)*a(n-1) + 24*(n-2)*a(n-2) - 168*(n-2)*a(n-3) )/(n+1). (End)

%t CoefficientList[Series[(1-Sqrt[1-24*x^2])/(12*x^2 -x*(1-Sqrt[1-24*x^2])), {x, 0, 40}], x] (* _G. C. Greubel_, Nov 07 2022 *)

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(1-24*x^2))/(12*x^2-x*(1-Sqrt(1-24*x^2))) )); // _G. C. Greubel_, Nov 07 2022

%o (SageMath)

%o def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)

%o def A132373(n): return sum(6^(n-k)*A120730(n,k) for k in range(n+1))

%o [A132373(n) for n in range(51)] # _G. C. Greubel_, Nov 07 2022

%Y Cf. A000012, A000108, A001405, A120730, A121724, A126087, A128386, A128387.

%K easy,nonn

%O 0,3

%A _Philippe Deléham_, Nov 10 2007

%E Terms beyond a(7) added by _R. J. Mathar_, Nov 19 2009