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Expansion of c(8*x^2)/(1 - x*c(8*x^2)), where c(x) is the g.f. of A000108.
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%I #9 Nov 09 2022 00:07:33

%S 1,1,9,17,153,353,3177,8113,73017,198401,1785609,5060433,45543897,

%T 133071009,1197639081,3581326065,32231934585,98156060225,883404542025,

%U 2730108129937,24570973169433,76862217117665,691759954058985

%N Expansion of c(8*x^2)/(1 - x*c(8*x^2)), where c(x) is the g.f. of A000108.

%C Hankel transform is 8^C(n+1, 2).

%C Series reversion of x*(1+x)/(1+2*x+9*x^2).

%H G. C. Greubel, <a href="/A132375/b132375.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..n} A120730(n,k) * 8^(n-k).

%F From _G. C. Greubel_, Nov 08 2022: (Start)

%F a(n) = (9*(n+1)*a(n-1) + 32*(n-2)*a(n-2) - 288*(n-2)*a(n-3))/(n+1).

%F G.f.: (1 - sqrt(1-32*x^2))/(16*x^2 - x*(1 - sqrt(1-32*x^2))). (End)

%t CoefficientList[Series[(1-Sqrt[1-32*x^2])/(16*x^2-x*(1-Sqrt[1-32*x^2])), {x,0, 40}], x] (* _G. C. Greubel_, Nov 08 2022 *)

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(1-32*x^2))/(16*x^2 -x*(1-Sqrt(1-32*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 A132375(n): return sum(8^(n-k)*A120730(n,k) for k in range(n+1))

%o [A132375(n) for n in range(51)] # _G. C. Greubel_, Nov 08 2022

%Y Cf. A001405, A126087, A128386, A121724, A128387, A121725.

%K easy,nonn

%O 0,3

%A _Philippe Deléham_, Nov 10 2007