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
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Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..200
Programs
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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 *)
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
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