A237928 Triangular array read by rows. T(n,k) is the number of n-permutations with k cycles of length one or k cycles of length two, n>=0,0<=k<=n.

1, 1, 1, 2, 1, 1, 3, 3, 0, 1, 18, 14, 9, 0, 1, 95, 75, 35, 10, 0, 1, 540, 369, 135, 55, 15, 0, 1, 3759, 2800, 1239, 420, 70, 21, 0, 1, 30310, 22980, 10570, 2884, 735, 112, 28, 0, 1, 272817, 202797, 87534, 24780, 6489, 1134, 168, 36, 0, 1

Offset: 0

Author

Geoffrey Critzer, Feb 15 2014

Keywords

• nonn
• tabl

Examples

			1,
1,    1,
2,    1,    1,
3,    3,    0,    1,
18,   14,   9,    0,   1,
95,   75,   35,   10,  0,  1,
540,  369,  135,  55,  15, 0,  1,
3759, 2800, 1239, 420, 70, 21, 0, 1
T(3,0)=3 because we have: (1)(2)(3);(1,2,3);(2,1,3)
		

Programs

Mathematica

nn=10;c=Sum[y^n x^(3n)/(2^n*n!^2),{n,0,nn}];Table[Take[(Range[0,nn]!CoefficientList[Series[Exp[y x]Exp[-x]/(1-x)+Exp[y x^2/2]Exp[-x^2/2]/(1-x)-c Exp[-x-x^2/2!]/(1-x),{x,0,nn}],{x,y}])[[n]],n],{n,1,nn}]//Grid

Formula

E.g.f.: A(x,y) + B(x,y) - C(x,y) where A(x,y) is e.g.f. for A008290, B(x,y) is e.g.f. for A114320, and C(x,y) = exp(-x - x^2/2)/(1-x)*Sum_{n>=0}y^n*x^(3n)/(2^n*n!^2).