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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 26 08:00 EDT 2021. Contains 348267 sequences. (Running on oeis4.)