%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