A237664 Interpolation polynomial through n+1 points (0,1), (1,1), ..., (n-1,1) and (n,n) evaluated at 2n.

0, 1, 7, 41, 211, 1009, 4621, 20593, 90091, 388961, 1662805, 7054321, 29745717, 124807201, 521515801, 2171645281, 9016205851, 37337699521, 154277300101, 636214748401, 2619084047581, 10765157488801, 44186078238121, 181135476007201, 741694884711301

Offset: 0

Author

Alois P. Heinz, Feb 11 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000290 (evaluated at n+1), A127736 (at n+2), A237622 (n points).

Programs

Maple

a:= proc(n) option remember; `if`(n<2, n,
       ((n-1)*(3*n-4)*(5*n-3) *a(n-1)
        -2*(2*n-3)*(3*n^2-4*n+2) *a(n-2))/
        (n*(3*n^2-10*n+9)))
    end:
seq(a(n), n=0..30);

Mathematica

CoefficientList[Series[(6*x-1)/Sqrt[1-4*x]^3-1/(x-1), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 14 2014 *)
a[n_] := Module[{m}, InterpolatingPolynomial[Table[{k, If[k == n, n, 1]}, {k, 0, n}], m] /. m -> 2n];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 22 2020 *)

Formula

G.f.: (6*x-1)/sqrt(1-4*x)^3 - 1/(x-1).

a(n) ~ sqrt(n)*4^n/sqrt(Pi). - Vaclav Kotesovec, Feb 14 2014

From Gregory Morse, Mar 19 2021: (Start)

a(n) = (2*n)!*(n-1)/(n!)^2 + 1.

a(n) = A030662(n-1)*(n-1) + n, for n > 0. (End)

E.g.f.: exp(x) * (1 - exp(x) * ((1 - 2*x) * BesselI(0,2*x) - 2 * x * BesselI(1,2*x))). - Ilya Gutkovskiy, Nov 19 2021