

A171998


In general, let A(n,k,m) denote the (n,k)th entry of the inverse of the matrix consisting of the (n,k)th mrestrained Stirling numbers of the second kind (1)^(nk) times the number of permutations of an nset with k disjoint cycles of length less than or equal to m) as the (n+1,k+1)th entry. The sequence shows A(n,k,3), which is a lower triangular matrix, read by rows.


1



1, 1, 1, 1, 3, 1, 5, 7, 6, 1, 65, 15, 25, 10, 1, 455, 455, 0, 65, 15, 1, 1295, 4725, 1715, 140, 140, 21, 1
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OFFSET

1,5


COMMENTS

A(n,k,m) also can be expanded for nonpositive integers n and k using the mrestrained Stirling numbers of the first kind.


LINKS

Table of n, a(n) for n=1..28.
Ji Young Choi, Multirestrained Stirling numbers, Ars Comb. 120 (2015), 113127.
John Engbers, David Galvin, and Cliff Smyth, Restricted Stirling and Lah number matrices and their inverses, Journal of Combinatorial Theory, Series A, 161 (2019), 271298.


FORMULA

A(n,k,m) = A(n1,k1,m)  Sum_{i=1..m1} (1)^{i}(k)...(k+i1) A(n,k+i,m) A(n,k,m) = A(n1,k1,m) + k A(n1,k,m) + (1)^m k(k+1)...(k+m1)A(n,k+m,m).


EXAMPLE

A(1,1,3) = 1, A(1,2,3) = 0, A(1,3,3) = 0, A(1,4,3) = 0, ...
A(2,1,3) = 1, A(2,2,3) = 1, A(2,3,3) = 0, A(2,4,3) = 0, ...
A(3,1,3) = 1, A(3,2,3) = 3, A(3,3,3) = 1, A(3,4,3) = 0, ...
A(4,1,3) = 5, A(4,2,3) = 7, A(4,3,3) = 6, A(4,4,3) = 1, ...
In other words, A(n,k,3) is the matrix
1
1 1
1 3 1
5 7 6 1
...
with all other entries in each row being 0.  N. J. A. Sloane, Dec 21 2019


CROSSREFS

Cf. A111246, A144633.
Sequence in context: A107920 A169998 A326729 * A159285 A021080 A049764
Adjacent sequences: A171995 A171996 A171997 * A171999 A172000 A172001


KEYWORD

sign,more,tabl


AUTHOR

Ji Young Choi, Jan 21 2010


STATUS

approved



