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
Stella Brassesco and Miguel A. Méndez, The asymptotic expansion for the factorial and Lagrange inversion formula, arXiv:1002.3894v1 [math.CA], 2010.
Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See p. 5.
Gergő Nemes, On the Coefficients of the Asymptotic Expansion of n!, J. Int. Seq. 13 (2010), 10.6.6.
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
KEYWORD
nonn,tabf
AUTHOR
STATUS
approved