A188143 Binomial transform of A187848.

1, 5, 29, 193, 1453, 12209, 113237, 1149241, 12675661, 151095569, 1937411429, 26614052617, 390244490749, 6087782363009, 100728768290645, 1762767028074937, 32542231109506285, 632202858036492593, 12895661952702667205

Offset: 0

Author

Groux Roland, Mar 24 2011

Keywords

• nonn

Comments

a(n) is also the INVERTi transform of A010842(n+1) starting at n=2.

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.

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).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000023.

Programs

Maple

with(LinearAlgebra):
c:= proc(n) option remember; add(n!/k!, k=0..n) end:
b:= n-> (-1)^(n+1) * Determinant(Matrix(n+2,
        (i, j)-> `if`(0<=i+1-j, c(i+1-j), 0))):
a:= proc(n) add(b(k) *binomial(n,k), k=0..n) end:
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 24 2011

Mathematica

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 *)

Formula

a(n) = Integral_{x=2..oo} x^n*exp(x-2)/((Ei(x-2))^2 + Pi^2) dx.

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

a(n) ~ exp(2) * n^2 * n!. - Vaclav Kotesovec, Nov 02 2023