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

0,2

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

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

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 *)

Crossrefs

Cf. A000023.

Sequence in context: A171267 A258314 A225030 * A095000 A086672 A324962

Adjacent sequences: A188140 A188141 A188142 * A188144 A188145 A188146

Keyword

nonn

Author

Groux Roland, Mar 24 2011

Status

approved