A363571 - OEIS

%I #8 Aug 24 2023 10:33:23

%S 0,1,8,61,480,3881,31976,266981,2251136,19124881,163452168,1403748941,

%T 12104113632,104723793721,908680775528,7904234296181,68905275700736,

%U 601832985410081,5265459181363976,46137526574521181,404821208100919520,3556361565584509001

%N Expansion of (1 / sqrt(1 - 10*x + 9*x^2) - 1 / (1 - x)) / 4.

%F E.g.f.: exp(x) * (exp(4*x) * BesselI(0,4*x) - 1) / 4.

%F a(n) = Sum_{k=1..n} binomial(n,k)^2 * 4^(k-1).

%F a(n) = (3^n * LegendreP(n,5/3) - 1) / 4.

%F a(n) = (A084771(n) - 1) / 4.

%t nmax = 21; CoefficientList[Series[(1/Sqrt[1 - 10 x + 9 x^2] - 1/(1 - x))/4, {x, 0, nmax}], x]

%t nmax = 21; CoefficientList[Series[Exp[x] (Exp[4 x] BesselI[0, 4 x] - 1)/4, {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[Binomial[n, k]^2 4^(k - 1), {k, 1, n}], {n, 0, 21}]

%t Table[(3^n LegendreP[n, 5/3] - 1)/4, {n, 0, 21}]

%Y Cf. A030662, A047665, A084771, A098665, A363570.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Aug 17 2023