

A143492


Unsigned 3Stirling numbers of the first kind.


7



1, 3, 1, 12, 7, 1, 60, 47, 12, 1, 360, 342, 119, 18, 1, 2520, 2754, 1175, 245, 25, 1, 20160, 24552, 12154, 3135, 445, 33, 1, 181440, 241128, 133938, 40369, 7140, 742, 42, 1, 1814400, 2592720, 1580508, 537628, 111769, 14560, 1162, 52, 1, 19958400
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OFFSET

3,2


COMMENTS

See A049458 for a signed version of this array. The unsigned 3Stirling numbers of the first kind count the permutations of the set {1,2,...,n} into k disjoint cycles, with the restriction that the elements 1, 2 and 3 belong to distinct cycles. This is the case r = 3 of the unsigned rStirling numbers of the first kind. For other cases see abs(A008275) (r = 1), A143491 (r = 2) and A143493 (r = 4). See A143495 for the corresponding 3Stirling numbers of the second kind. The theory of rStirling numbers of both kinds is developed in [Broder]. For details of the related 3Lah numbers see A143498.
With offset n=0 and k=0, this is the Sheffer triangle (1/(1x)^3, log(1x)) (in the umbral notation of S. Roman's book this would be called Sheffer for (exp(3*t), 1exp(t))). See the e.g.f given below. Compare also with the e.g.f. for the signed version A049458.  Wolfdieter Lang, Oct 10 2011
With offset n=0 and k=0 : triangle T(n,k), read by rows, given by (3,1,4,2,5,3,6,4,7,5,8,6,...) DELTA (1,0,1,0,1,0,1,0,1,0,1,...) where DELTA is the operator defined in A084938.  Philippe Deléham, Oct 31 2011


LINKS

Table of n, a(n) for n=3..48.
Broder Andrei Z., The rStirling numbers, Discrete Math. 49, 241259 (1984)
A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1, 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2.  N. J. A. Sloane, Mar 28 2015]
Askar Dzhumadil’daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 No. 13, 3351 (2001).
Michael J. Schlosser and Meesue Yoo, Elliptic Rook and File Numbers, Electronic Journal of Combinatorics, 24(1) (2017), #P1.31.


FORMULA

T(n,k) = (n3)! * Sum_{j = k3 .. n3} C(nj1,2)*Stirling1(j,k3)/j!.
Recurrence relation: T(n,k) = T(n1,k1) + (n1)*T(n1,k) for n > 3, with boundary conditions: T(n,2) = T(2,n) = 0, for all n; T(3,3) = 1; T(3,k) = 0 for k > 3.
Special cases:
T(n,3) = (n1)!/2! for n >= 3.
T(n,4) = (n1)!/2!*(1/3 + ... + 1/(n1)) for n >= 3.
T(n,k) = Sum_{3 <= i_1 < ... < i_(nk) < n} (i_1*i_2* ...*i_(nk)). For example, T(6,4) = Sum_{3 <= i < j < 6} (i*j) = 3*4 + 3*5 + 4*5 = 47.
Row g.f.: Sum_{k = 3..n} T(n,k)*x^k = x^3*(x+3)*(x+4)* ... *(x+n1).
E.g.f. for column (k+3): Sum_{n = k..inf} T(n+3,k+3)*x^n/n! = 1/k!*1/(1x)^3 * (log(1/(1x)))^k.
E.g.f.: (1/(1t))^(x+3) = Sum_{n = 0..inf} Sum_{k = 0..n} T(n+3,k+3)*x^k*t^n/n! = 1 + (3+x)*t/1! + (12+7*x+x^2)*t^2/2! + ....
This array is the matrix product St1 * P^2, where St1 denotes the lower triangular array of unsigned Stirling numbers of the first kind, abs(A008275) and P denotes Pascal's triangle, A007318. The row sums are n!/3! ( A001715 ). The alternating row sums are (n2)!.
If we define f(n,i,a) = sum(binomial(n,k)*Stirling1(nk,i)*product(aj,j=0..k1),k=0..ni), then T(n,i) = f(n,i,3), for n=1,2,...;i=0...n.  Milan Janjic, Dec 21 2008


EXAMPLE

Triangle begins
n\k.....3.....4.....5.....6.....7.....8
========================================
3.......1
4.......3.....1
5......12.....7.....1
6......60....47....12.....1
7.....360...342...119....18.....1
8....2520..2754..1175...245....25.....1
...
T(5,4) = 7. The permutations of {1,2,3,4,5} with 4 cycles such that 1, 2 and 3 belong to different cycles are: (14)(2)(3)(5), (15)(2)(3)(4), (24)(1)(3)(5), (25)(1)(3)(4), (34)(1)(2)(5), (35)(1)(2)(4) and (45)(1)(2)(3).


MAPLE

with combinat: T := (n, k) > (n3)! * add(binomial(nj1, 2)*abs(stirling1(j, k3))/j!, j = k3..n3): for n from 3 to 12 do seq(T(n, k), k = 3..n) end do;


CROSSREFS

Cf. A001710  A001714 (column 3  column 7), A001715 (row sums), A008275, A049458 (signed version), A143491, A143493, A143495, A143498.
Sequence in context: A258245 A133366 A049458 * A243662 A062139 A156366
Adjacent sequences: A143489 A143490 A143491 * A143493 A143494 A143495


KEYWORD

easy,nonn,tabl


AUTHOR

Peter Bala, Aug 20 2008


STATUS

approved



