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.

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

Table of n, a(n) for n=4..63.

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

Cf. A008284, A245182, A245183.

Sequence in context: A362824 A213268 A291118 * A284414 A355614 A140274

Adjacent sequences: A245181 A245182 A245183 * A245185 A245186 A245187

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