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A245184 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, 4) into exactly k disjoint cycles. 2

%I

%S 1,2,1,1,1,4,4,2,1,5,15,17,10,4,1,30,85,97,61,25,7,1,210,595,691,451,

%T 192,57,11,1,1680,4844,5778,3904,1724,536,117,16,1,15120,44604,54846,

%U 38396,17593,5700,1339,219,22,1,151200,456840,579360,420258,199870,67361,16709,3032,380,29,1

%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, 4) 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 = 4 for n >= r. - _Andrew Howroyd_, Feb 24 2020

%e Triangle begins:

%e 1 2 1 1

%e 1 4 4 2 1

%e 5 15 17 10 4 1

%e 30 85 97 61 25 7 1

%e 210 595 691 451 192 57 11 1

%e ...

%o T(n)={my(r=4, 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=4, #A, print(A[n, 1..n])) } \\ _Andrew Howroyd_, Feb 24 2020

%Y Cf. A008284, A245182, A245183.

%K nonn,tabf

%O 4,2

%A _N. J. A. Sloane_, Jul 17 2014

%E Terms a(34) and beyond from _Andrew Howroyd_, Feb 24 2020

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Last modified October 18 23:24 EDT 2021. Contains 348070 sequences. (Running on oeis4.)