%I #9 Aug 24 2023 10:33:12

%S 0,1,7,45,295,1981,13545,93829,656311,4625181,32788657,233567269,

%T 1670457321,11987269477,86268665917,622391877045,4500029549911,

%U 32598283556317,236542093805025,1719008077215205,12509403045819505,91143878730342021,664816240262272237

%N Expansion of (1 / sqrt(1 - 8*x + 4*x^2) - 1 / (1 - x)) / 3.

%F E.g.f.: exp(x) * (exp(3*x) * BesselI(0,2*sqrt(3)*x) - 1) / 3.

%F a(n) = Sum_{k=1..n} binomial(n,k)^2 * 3^(k-1).

%F a(n) = (2^n * LegendreP(n,2) - 1) / 3.

%F a(n) = (A069835(n) - 1) / 3.

%t nmax = 22; CoefficientList[Series[(1/Sqrt[1 - 8 x + 4 x^2] - 1/(1 - x))/3, {x, 0, nmax}], x]

%t nmax = 22; CoefficientList[Series[Exp[x] (Exp[3 x] BesselI[0, 2 Sqrt[3] x] - 1)/3, {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[Binomial[n, k]^2 3^(k - 1), {k, 1, n}], {n, 0, 22}]

%t Table[(2^n LegendreP[n, 2] - 1)/3, {n, 0, 22}]

%Y Cf. A030662, A047665, A069835, A098663, A363571.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Aug 17 2023