A363570 Expansion of (1 / sqrt(1 - 8*x + 4*x^2) - 1 / (1 - x)) / 3.

0, 1, 7, 45, 295, 1981, 13545, 93829, 656311, 4625181, 32788657, 233567269, 1670457321, 11987269477, 86268665917, 622391877045, 4500029549911, 32598283556317, 236542093805025, 1719008077215205, 12509403045819505, 91143878730342021, 664816240262272237

Offset: 0

Author

Ilya Gutkovskiy, Aug 17 2023

Keywords

• nonn

Crossrefs

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

Programs

Mathematica

nmax = 22; CoefficientList[Series[(1/Sqrt[1 - 8 x + 4 x^2] - 1/(1 - x))/3, {x, 0, nmax}], x]
nmax = 22; CoefficientList[Series[Exp[x] (Exp[3 x] BesselI[0, 2 Sqrt[3] x] - 1)/3, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k]^2 3^(k - 1), {k, 1, n}], {n, 0, 22}]
Table[(2^n LegendreP[n, 2] - 1)/3, {n, 0, 22}]

Formula

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

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

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

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