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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
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, 192, 57, 11, 1, 1680, 4844, 5778, 3904, 1724, 536, 117, 16, 1, 15120, 44604, 54846, 38396, 17593, 5700, 1339, 219, 22, 1, 151200, 456840, 579360, 420258, 199870, 67361, 16709, 3032, 380, 29, 1
OFFSET
4,2
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-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
EXAMPLE
Triangle begins:
1 2 1 1
1 4 4 2 1
5 15 17 10 4 1
30 85 97 61 25 7 1
210 595 691 451 192 57 11 1
...
PROG
(PARI)
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]))))}
{ my(A=T(10)); for(n=4, #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(34) and beyond from Andrew Howroyd, Feb 24 2020
STATUS
approved