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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.
0
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
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
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Feb 15 2014
STATUS
approved