A245183 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.

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, 2254, 2419, 1409, 505, 116, 16, 1, 6720, 18704, 21112, 13098, 5069, 1296, 218, 22, 1, 60480, 174096, 205680, 135036, 55916, 15568, 2975, 379, 29, 1

Offset

3,5

Links

Table of n, a(n) for n=3..54.

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 = 3 for n >= r. - Andrew Howroyd, Feb 24 2020

Example

Triangle begins:
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 2254 2419 1409 505 116 16 1
...

Prog

(PARI)
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]))))}
{ my(A=T(10)); for(n=3, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Feb 24 2020

Crossrefs

Cf. A008284, A245182, A245184.

Sequence in context: A119263 A028412 A156699 * A262347 A182236 A077819

Adjacent sequences: A245180 A245181 A245182 * A245184 A245185 A245186

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