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%I #12 Feb 24 2020 17:16:09
%S 1,1,1,1,3,2,1,4,10,9,4,1,20,50,48,24,7,1,120,310,315,171,56,11,1,840,
%T 2254,2419,1409,505,116,16,1,6720,18704,21112,13098,5069,1296,218,22,
%U 1,60480,174096,205680,135036,55916,15568,2975,379,29,1
%N Irregular triangle read by rows: T(n,k) (n>=2, 1<=k<=n) gives number of arrangements of the elements from the multiset M(n, 3) into exactly k disjoint cycles.
%H Martin Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Griffiths/griffiths31.html">Generating Functions for Extended Stirling Numbers of the First Kind</a>, Journal of Integer Sequences, 17 (2014), #14.6.4.
%F T(n,k) = Sum_{m=0..k} |Stirling1(n-r, m)| * Sum_{j=0, r-k+m} binomial(n+j-r-1, j) * A008284(r-j, k-m) where r = 3 for n >= r. - _Andrew Howroyd_, Feb 24 2020
%e Triangle begins:
%e 1 1 1
%e 1 3 2 1
%e 4 10 9 4 1
%e 20 50 48 24 7 1
%e 120 310 315 171 56 11 1
%e 840 2254 2419 1409 505 116 16 1
%e ...
%o (PARI)
%o T(n)={my(r=3, P=matrix(1+r, 1+r, n, k, #partitions(n-k, k-1))); matrix(n, n, n, k, if(n<r, 0, sum(m=0, k, abs(stirling(n-r, m, 1)) * sum(j=0, r-k+m, binomial(n+j-r-1, j)*P[1+r-j, 1+k-m]))))}
%o { my(A=T(10)); for(n=3, #A, print(A[n, 1..n])) } \\ _Andrew Howroyd_, Feb 24 2020
%Y Cf. A008284, A245182, A245184.
%K nonn,tabf
%O 3,5
%A _N. J. A. Sloane_, Jul 17 2014
%E Terms a(34) and beyond from _Andrew Howroyd_, Feb 24 2020
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