OFFSET
2,4
COMMENTS
The multiset M(n, 2) consists of 2 1's and one copy of each of 2..n-1. For example, M(5,2) is the multiset {1,1,2,3,4}. See Griffiths reference for formula. - Andrew Howroyd, Feb 24 2020
LINKS
Martin Griffiths, Generating Functions for Extended Stirling Numbers of the First Kind, Journal of Integer Sequences, 17 (2014), #14.6.4.
FORMULA
T(n,k) = Sum_{m=0..k} |Stirling1(n-2, m)| * Sum_{j=0, 2-k+m} binomial(n+j-3, j) * A008284(2-j, k-m) for n >= 2. - Andrew Howroyd, Feb 24 2020
EXAMPLE
Triangle begins:
1 1
1 2 1
3 6 4 1
12 26 20 7 1
60 140 121 51 11 1
360 894 849 410 110 16 1
2520 6594 6763 3634 1135 211 22 1
...
From Andrew Howroyd, Feb 24 2020: (Start)
For n = 4, arrangements of the multiset {1,1,2,3} into cycles are:
T(4,1) = 3: (1123), (1132), (1213);
T(4,2) = 6: (112)(3), (113)(2), (123)(1), (132)(1), (11)(23) (12)(13);
T(4,3) = 4: (11)(2)(3), (12)(1)(3), (13)(1)(2), (23)(1)(1);
T(4,4) = 1: (1)(1)(2)(3).
(End)
PROG
(PARI)
T(n)={my(P=matrix(3, 3, n, k, #partitions(n-k, k-1))); matrix(n, n, n, k, if(n<2, 0, sum(m=0, k, abs(stirling(n-2, m, 1)) * sum(j=0, 2-k+m, binomial(n+j-3, j)*P[1+2-j, 1+k-m]))))}
{my(A=T(10)); for(n=2, #A, print(A[n, 1..n]))} \\ Andrew Howroyd, Feb 24 2020
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Jul 17 2014
EXTENSIONS
Terms a(36) and beyond from Andrew Howroyd, Feb 24 2020
STATUS
approved