A187848 a(n) is the moment of order n for the probability density function defined by rho(x)=exp(x-1)/((Ei(x-1))^2+Pi^2) over the interval 1..infinity, with Ei the exponential integral.

1, 4, 20, 120, 836, 6608, 58324, 568296, 6060340, 70245856, 879937892, 11853424536, 170963881892, 2629912684784, 42995842035316, 744683072665416, 13624184625098644, 262594854417561856, 5319099368762699012, 112977659152942035192, 2511041582408699358980

Offset

0,2

Comments

a(n) is also the binomial transform of A003319(n+1).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Formula

Let c(n)=A000522(n) and An the square matrix of order n+2 defined by: if j<=i A[i,j]=c(i-j+1); A(i,i+1)=1; if j>i+1 A[i,j]=0; then a(n)=(-1)^(n+1)*det(An).

G.f.: (1 - 2*x - U(0))/x^2 where U(k)= 1 - x - x*(k+1)/(1 - x*(k+1)/U(k+1)) ; (continued fraction, 2-step). - Sergei N. Gladkovskii, Oct 14 2012

G.f.: 1/x^2 - 1/x -1/(x^2*W(0)) , where W(k) = 1 - x*(k+1)/( x*(k+1) - 1/(1 - x*(k+1)/( x*(k+2) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 25 2013

G.f.: W(0)/x - 1/x, where W(k) = 1 - x*(k+1)/( x*(k+2) - 1/(1 - x*(k+1+R)/( x*(k+1+R) - 1/W(k+1) ))); R=1 (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013

a(n) ~ exp(1) * n! * n^2 * (1 - 1/n - 4/n^3 - 23/n^4 - 175/n^5 - 1615/n^6 - 17375/n^7 - 212607/n^8 - 2909007/n^9 - 43953071/n^10). - Vaclav Kotesovec, Sep 02 2014, updated Aug 01 2015

Maple

with(LinearAlgebra):
c:= proc(n) option remember; add(n!/k!, k=0..n) end:
a:= n-> (-1)^(n+1) *Determinant(Matrix(n+2,
        (i, j)-> `if`(0<=i+1-j, c(i+1-j), 0))):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 24 2011
# second Maple program:
b:= proc(n) option remember;
      `if`(n<0, -1, -add(b(n-i)*i!, i=1..n+1))
    end:
a:= n-> add(b(k+1)*binomial(n, k), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 26 2013

Mathematica

b[n_] := b[n] = If[n<0, -1, -Sum[b[n-i]*i!, {i, 1, n+1}]]; a[n_] := Sum[b[k+1] * Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 30 2015, after Alois P. Heinz *)

Crossrefs

Sequence in context: A277920 A093123 A092055 * A370653 A001715 A304069

Adjacent sequences: A187845 A187846 A187847 * A187849 A187850 A187851

Keyword

nonn

Author

Groux Roland, Mar 14 2011

Status

approved