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%I #18 Nov 09 2022 00:07:09
%S 1,1,8,15,120,274,2192,5531,44248,118686,949488,2654646,21237168,
%T 61189668,489517344,1443039123,11544312984,34648845862,277190766896,
%U 844131474530,6753051796240,20813234394492,166505875155936,518373091849502
%N Expansion of c(7*x^2)/(1 - x*c(7*x^2)), where c(x) is the g.f. of A000108.
%C Hankel transform is 7^C(n+1,2).
%C Series reversion of x*(1+x)*(1+2*x+8*x^2).
%H G. C. Greubel, <a href="/A132374/b132374.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{k=0..n} A120730(n,k) * 7^(n-k).
%F From _G. C. Greubel_, Nov 08 2022: (Start)
%F a(n) = 4*( 2*(n+1)*a(n-1) + 7*(n-2)*a(n-2) - 56*(n-2)*a(n-3) )/(n+1).
%F G.f.: (1 - sqrt(1 - 28*x^2))/(14*x^2 - x*(1 - sqrt(1 - 28*x^2))). (End)
%t CoefficientList[Series[(1-Sqrt[1-28*x^2])/(14*x^2 -x*(1-Sqrt[1-28*x^2])), {x,0,40}], x] (* _G. C. Greubel_, Nov 08 2022 *)
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(1-28*x^2))/(14*x^2 -x*(1-Sqrt(1-28*x^2))) )); // _G. C. Greubel_, Nov 08 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 A132374(n): return sum(7^(n-k)*A120730(n,k) for k in range(n+1))
%o [A132374(n) for n in range(51)] # _G. C. Greubel_, Nov 08 2022
%Y Cf. A000108, A001405, A120730, A126087, A128386, A121724, A128387, A121725.
%K easy,nonn
%O 0,3
%A _Philippe Deléham_, Nov 10 2007
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