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A237622 Interpolation polynomial through n points (0,1), (1,1), ..., (n-2,1) and (n-1,n) evaluated at 2n, a(0)=1.

%I A237622 #16 Dec 22 2020 06:41:34
%S A237622 1,1,5,31,169,841,3961,18019,80081,350065,1511641,6466461,27457585,
%T A237622 115892401,486748081,2035917451,8485840801,35263382881,146157442201,
%U A237622 604404010981,2494365759601,10275832148401,42264944401681,173588164506901,712027089322849
%N A237622 Interpolation polynomial through n points (0,1), (1,1), ..., (n-2,1) and (n-1,n) evaluated at 2n, a(0)=1.
%H A237622 Alois P. Heinz, <a href="/A237622/b237622.txt">Table of n, a(n) for n = 0..1000</a>
%F A237622 E.g.f.: exp(x)+2*exp(2*x)*(BesselI(1,2*x)*(x-1)+x*BesselI(0,2*x)).
%F A237622 a(n) ~ sqrt(n)*4^n/sqrt(Pi). - _Vaclav Kotesovec_, Feb 14 2014
%p A237622 a:= proc(n) option remember; `if`(n<3, [1, 1, 5][n+1],
%p A237622        (n*(15*n^3-44*n^2+43*n-18) *a(n-1)
%p A237622         -2*(n-1)*(2*n-3)*(3*n^2-n+2) *a(n-2))/
%p A237622         ((n-2)*(n+1)*(3*n^2-7*n+6)))
%p A237622     end:
%p A237622 seq(a(n), n=0..30);
%t A237622 a[n_] := Module[{m}, If[n == 0, 1, InterpolatingPolynomial[Table[{k, If[k == n-1, n, 1]}, {k, 0, n-1}], m] /. m -> 2n]];
%t A237622 a /@ Range[0, 30] (* _Jean-François Alcover_, Dec 22 2020 *)
%Y A237622 Cf. A002061 (evaluated at n), A158842 (at n+1), A237664 (n+1 points).
%K A237622 nonn
%O A237622 0,3
%A A237622 _Alois P. Heinz_, Feb 10 2014