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

0,4

Links

Table of n, a(n) for n=0..54.

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).

Example

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)

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

Crossrefs

Sequence in context: A078802 A216232 A217765 * A108482 A124750 A275865

Adjacent sequences: A237925 A237926 A237927 * A237929 A237930 A237931

Keyword

nonn,tabl

Author

Geoffrey Critzer, Feb 15 2014

Status

approved