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

%I A237664 #28 Nov 19 2021 11:08:37
%S A237664 0,1,7,41,211,1009,4621,20593,90091,388961,1662805,7054321,29745717,
%T A237664 124807201,521515801,2171645281,9016205851,37337699521,154277300101,
%U A237664 636214748401,2619084047581,10765157488801,44186078238121,181135476007201,741694884711301
%N A237664 Interpolation polynomial through n+1 points (0,1), (1,1), ..., (n-1,1) and (n,n) evaluated at 2n.
%H A237664 Alois P. Heinz, <a href="/A237664/b237664.txt">Table of n, a(n) for n = 0..1000</a>
%F A237664 G.f.: (6*x-1)/sqrt(1-4*x)^3 - 1/(x-1).
%F A237664 a(n) ~ sqrt(n)*4^n/sqrt(Pi). - _Vaclav Kotesovec_, Feb 14 2014
%F A237664 From _Gregory Morse_, Mar 19 2021: (Start)
%F A237664 a(n) = (2*n)!*(n-1)/(n!)^2 + 1.
%F A237664 a(n) = A030662(n-1)*(n-1) + n, for n > 0. (End)
%F A237664 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
%p A237664 a:= proc(n) option remember; `if`(n<2, n,
%p A237664        ((n-1)*(3*n-4)*(5*n-3) *a(n-1)
%p A237664         -2*(2*n-3)*(3*n^2-4*n+2) *a(n-2))/
%p A237664         (n*(3*n^2-10*n+9)))
%p A237664     end:
%p A237664 seq(a(n), n=0..30);
%t A237664 CoefficientList[Series[(6*x-1)/Sqrt[1-4*x]^3-1/(x-1), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 14 2014 *)
%t A237664 a[n_] := Module[{m}, InterpolatingPolynomial[Table[{k, If[k == n, n, 1]}, {k, 0, n}], m] /. m -> 2n];
%t A237664 a /@ Range[0, 30] (* _Jean-François Alcover_, Dec 22 2020 *)
%Y A237664 Cf. A000290 (evaluated at n+1), A127736 (at n+2), A237622 (n points).
%K A237664 nonn
%O A237664 0,3
%A A237664 _Alois P. Heinz_, Feb 11 2014