OFFSET
0,4
COMMENTS
The derangement characterization would yield a(0) = 1, but -1 is the value given in Martin and Kearney's paper. - Eric M. Schmidt, Jul 10 2015
LINKS
Eric M. Schmidt, Table of n, a(n) for n = 0..300
Jesse Elliott, Asymptotic expansions of the prime counting function, arXiv:1809.06633 [math.NT], 2018.
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.
FORMULA
Martin and Kearney (2015) give both a recurrence and a g.f.
The recurrence is a(0)=-1, a(1)=0; a(n) = (n-1)*a(n-1) + (n-3)*a(n-2) + Sum_{j=1..n-1} a(j)*a(n-j).
a(n) ~ n!/exp(1) * (1 - 2/n^2 - 6/n^3 - 29/n^4 - 196/n^5 - 1665/n^6 - 16796/n^7 - 194905/n^8 - 2549468/n^9 - 37055681/n^10), for coefficients see A260578. - Vaclav Kotesovec, Jul 28 2015
G.f.: -1 + x^2/(1 - 2*x - 4*x^2/(1 - 4*x - 9*x^2/(1 - 6*x - 16*x^2/(1 - ...)))), a continued fraction. - Ilya Gutkovskiy, Aug 22 2018
EXAMPLE
There are 9 derangements of 1,2,3,4. All of them are indecomposable except for 2,1,4,3. Thus a(4) = 8. - Eric M. Schmidt, Jul 10 2015
MATHEMATICA
Clear[a]; a[0]=-1; a[1]=0; a[n_]:=a[n]=(n-1)*a[n-1] + (n-3)*a[n-2] + Sum[a[j]*a[n-j], {j, 1, n-1}]; Table[a[n], {n, 0, 20}] (* Vaclav Kotesovec, Jul 29 2015 *)
nmax = 25; CoefficientList[Assuming[Element[x, Reals], Series[-x*E^(1 + 1/x)/ExpIntegralEi[1 + 1/x], {x, 0, nmax}]], x] (* Vaclav Kotesovec, Aug 05 2015 *)
PROG
(Sage)
def a(n) : return -1 if n==0 else 0 if n==1 else (n-1)*a(n-1) + (n-3)*a(n-2) + sum(a(j)*a(n-j) for j in [1..n-1]) # Eric M. Schmidt, Jul 10 2015
CROSSREFS
KEYWORD
sign,easy
AUTHOR
N. J. A. Sloane, Jul 09 2015
EXTENSIONS
More terms from and definition edited by Eric M. Schmidt, Jul 10 2015
STATUS
approved