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

%I #22 Sep 24 2013 15:31:17

%S 0,1,5,29,200,1609,14809,153453,1767240,22383681,309123733,4621295117,

%T 74331184256,1279614456041,23470211031097,456836915073277,

%U 9403557603534960,204061142480099649,4655419598313230277,111378768040665868093,2788108620329147151896

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

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

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

%F 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

%F 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

%t 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]

%Y Cf. A002720, A021009.

%K nonn

%O 0,3

%A _Geoffrey Critzer_, Sep 04 2012