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%I #26 Feb 10 2018 06:38:18
%S 2,6,24,120,40,720,420,5040,3948,40320,38304,2240,362880,396576,50400,
%T 3628800,4419360,859320,39916800,53048160,13665960,246400,479001600,
%U 684478080,216339552,9609600,6227020800,9464307840,3501834336
%N Triangle of number of permutations of {1, 2, ..., n} having exactly k cycles, each of which is of length >=r for r=3.
%C Generalizes Stirling numbers of the first kind
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 257.
%H Alois P. Heinz, <a href="/A050211/b050211.txt">Rows n = 3..200, flattened</a>
%H S. Brassesco, M. A. Méndez, <a href="http://arxiv.org/abs/1002.3894">The asymptotic expansion for the factorial and Lagrange inversion formula</a>, arXiv:1002.3894v1 [math.CA], 2010.
%H G. Nemes, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Nemes/nemes2.html">On the Coefficients of the Asymptotic Expansion of n!</a>, J. Int. Seq. 13 (2010), 10.6.6.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PermutationCycle.html">Permutation Cycle.</a>
%F From _Peter Bala_, Sep 06 2011: (Start)
%F 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!+....
%F E.g.f. for column k: 1/k!*(-log(1-x)-x-x^2/2)^k.
%F Starting at row 3, row lengths are 1, 1, 1, 2, 2, 2, 3, 3, 3, ....
%F Recurrence: T(n,k) = (n-1)*T(n-1,k) + (n-1)*(n-2)*T(n-3,k-1).
%F [End]
%e Table begins
%e .n\k.|......u.....u^2....u^3
%e = = = = = = = = = = = = = = =
%e ..3..|......2
%e ..4..|......6
%e ..5..|.....24
%e ..6..|....120.....40
%e ..7..|....720....420
%e ..8..|...5040...3948
%e ..9..|..40320..38304....2240
%e ..
%p b:= proc(n) option remember; expand(`if`(n=0, 1, add(
%p b(n-i)*x*binomial(n-1, i-1)*(i-1)!, i=3..n)))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
%p seq(T(n), n=3..15); # _Alois P. Heinz_, Sep 25 2016
%t 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_ *)
%Y Cf. A008275, A008306, A050212, A050213.
%K nonn,tabf
%O 3,1
%A _Eric W. Weisstein_
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