OFFSET
0,3
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.
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Aug 17 2023
STATUS
approved