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A188143 Binomial transform of A187848.

%I A188143 #36 Nov 02 2023 11:10:28
%S A188143 1,5,29,193,1453,12209,113237,1149241,12675661,151095569,1937411429,
%T A188143 26614052617,390244490749,6087782363009,100728768290645,
%U A188143 1762767028074937,32542231109506285,632202858036492593,12895661952702667205
%N A188143 Binomial transform of A187848.
%C A188143 a(n) is also the INVERTi transform of A010842(n+1) starting at n=2.
%C A188143 a(n) is also the moment of order n for the measure of density: exp(x-2) / ((Ei(x-2))^2+Pi^2) over the interval 2..infinity with Ei the exponential integral.
%C A188143 More generally, for every integer k, the sequence a(n,k)=int(x^n*exp(x-k) / ((Ei(x-k))^2+Pi^2), x=k..infinity) is the INVERTi transform of the sequence b(n+1,k), starting at n=2, with b(n,k)=int(x^n*exp(x-k), x=k..infinity) whose e.g.f. is exp(k*x)/(1-x).
%H A188143 Vaclav Kotesovec, <a href="/A188143/b188143.txt">Table of n, a(n) for n = 0..200</a>
%F A188143 a(n) = Integral_{x=2..oo} x^n*exp(x-2)/((Ei(x-2))^2 + Pi^2) dx.
%F A188143 G.f.: 1/x^2 - 3/x - Q(0)/x^2, where Q(k) = 1 - 2*x - x*(k+1)/(1-x*(k+1)/Q(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Apr 19 2013
%F A188143 a(n) ~ exp(2) * n^2 * n!. - _Vaclav Kotesovec_, Nov 02 2023
%p A188143 with(LinearAlgebra):
%p A188143 c:= proc(n) option remember; add(n!/k!, k=0..n) end:
%p A188143 b:= n-> (-1)^(n+1) * Determinant(Matrix(n+2,
%p A188143         (i, j)-> `if`(0<=i+1-j, c(i+1-j), 0))):
%p A188143 a:= proc(n) add(b(k) *binomial(n,k), k=0..n) end:
%p A188143 seq(a(n), n=0..20);  # _Alois P. Heinz_, Mar 24 2011
%t A188143 c[n_] := c[n] = Sum[n!/k!, {k, 0, n}]; b[n_] := (-1)^(n+1)*Det[Table[If[0 <= i+1-j, c[i+1-j], 0], {i, 1, n+2}, {j, 1, n+2}]]; a[n_] := Sum[b[k] * Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Feb 28 2017, after _Alois P. Heinz_ *)
%Y A188143 Cf. A000023.
%K A188143 nonn
%O A188143 0,2
%A A188143 _Groux Roland_, Mar 24 2011