A216313 Total number of cycles in all partial permutations of {1,2,...,n}.

0, 1, 5, 29, 200, 1609, 14809, 153453, 1767240, 22383681, 309123733, 4621295117, 74331184256, 1279614456041, 23470211031097, 456836915073277, 9403557603534960, 204061142480099649, 4655419598313230277, 111378768040665868093, 2788108620329147151896

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

E.g.f.: exp(x/(1-x))*log(1/(1-x))/(1-x).

a(n) = sum(k=0..n, A216294(n,k)*k ).

a(n) = (4*n-3)*a(n-1) - (6*n^2 - 17*n + 13)*a(n-2) + (n-2)^2*(4*n-9)*a(n-3) - (n-3)^3*(n-2)*a(n-4). - Vaclav Kotesovec, Sep 24 2013

a(n) ~ sqrt(2)/4 * n^(n+1/4) * exp(2*sqrt(n)-n-1/2) * (log(n)*(1 + 31/(48*sqrt(n)) + 553/(4608*n)) + 1/sqrt(n) + 43/(48*n)). - Vaclav Kotesovec, Sep 24 2013

Mathematica

nn=20; t=Sum[n^(n-1)x^n/n!, {n, 1, nn}]; Range[0, nn]!CoefficientList[ Series[D[Exp[ x/(1-x)]/(1-x)^y, y]/.y->1, {x, 0, nn}], x]

Crossrefs

Cf. A002720, A021009.

Sequence in context: A199419 A094710 A367050 * A108453 A201203 A004213

Adjacent sequences: A216310 A216311 A216312 * A216314 A216315 A216316

Keyword

nonn

Author

Geoffrey Critzer, Sep 04 2012

Status

approved