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A050211 Triangle of number of permutations of {1, 2, ..., n} having exactly k cycles, each of which is of length >=r for r=3. 3
2, 6, 24, 120, 40, 720, 420, 5040, 3948, 40320, 38304, 2240, 362880, 396576, 50400, 3628800, 4419360, 859320, 39916800, 53048160, 13665960, 246400, 479001600, 684478080, 216339552, 9609600, 6227020800, 9464307840, 3501834336 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,1

COMMENTS

Generalizes Stirling numbers of the first kind

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 257.

LINKS

Alois P. Heinz, Rows n = 3..200, flattened

S. Brassesco, M. A. Méndez, The asymptotic expansion for the factorial and Lagrange inversion formula, arXiv:1002.3894v1 [math.CA]

Eric Weisstein's World of Mathematics, Permutation Cycle.

FORMULA

From Peter Bala, Sep 06 2011: (Start)

E.g.f.: (1-t)^(-u)*exp(-u*(t+t^2/2)) - 1 = (2*u)*t^3/3!+(6*u)*t^4/4!+(24*u)*t^5/5!+(120*u+40*u^2)*t^6/6!+(720*u+420*u^2)*t^7/7!+....

E.g.f. for column k: 1/k!*(-log(1-x)-x-x^2/2)^k.

Starting at row 3, row lengths are 1, 1, 1, 2, 2, 2, 3, 3, 3, ....

Recurrence: T(n,k) = (n-1)*T(n-1,k) + (n-1)*(n-2)*T(n-3,k-1).

[End]

EXAMPLE

Table begins

.n\k.|......u.....u^2....u^3

= = = = = = = = = = = = = = =

..3..|......2

..4..|......6

..5..|.....24

..6..|....120.....40

..7..|....720....420

..8..|...5040...3948

..9..|..40320..38304....2240

..

MAPLE

b:= proc(n) option remember; expand(`if`(n=0, 1, add(

      b(n-i)*x*binomial(n-1, i-1)*(i-1)!, i=3..n)))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):

seq(T(n), n=3..15);  # Alois P. Heinz, Sep 25 2016

MATHEMATICA

t[n_ /; n >= 3, k_ /; k >= 1] := t[n, k] = (n - 1)*t[n - 1, k] + (n - 2)*(n - 1)*t[n - 3, k - 1] ; t[_, _] = 0; t[3, 1] = 2; Flatten[ Table[t[n, k], {n, 3, 15}, {k, 1, Floor[n/3]}]] (* Jean-François Alcover, Nov 05 2012, after Peter Bala *)

CROSSREFS

Cf. A008275, A008306, A050212, A050213.

Sequence in context: A260850 A008336 A033643 * A248766 A263713 A242427

Adjacent sequences:  A050208 A050209 A050210 * A050212 A050213 A050214

KEYWORD

nonn,tabf

AUTHOR

Eric W. Weisstein

STATUS

approved

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Last modified December 7 11:24 EST 2016. Contains 278874 sequences.