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A059022 Triangle of Stirling numbers of order 3. 6
1, 1, 1, 1, 10, 1, 35, 1, 91, 1, 210, 280, 1, 456, 2100, 1, 957, 10395, 1, 1969, 42735, 15400, 1, 4004, 158301, 200200, 1, 8086, 549549, 1611610, 1, 16263, 1827826, 10335325, 1401400, 1, 32631, 5903898, 57962905, 28028000, 1, 65382, 18682014 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,5

COMMENTS

The number of partitions of the set N, |N|=n, into k blocks, all of cardinality greater than or equal to 3. This is the 3-associated Stirling number of the second kind (Comtet) or the Stirling number of order 3 (Fekete).

This is entered as a triangular array. The entries S_3(n,k) are zero for 3k>n, so these values are omitted. Initial entry in sequence is S_3(3,1).

Rows are of lengths 1,1,1,2,2,2,3,3,3,...

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.

LINKS

Table of n, a(n) for n=3..45.

A. E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.

G. Nemes, On the Coefficients of the Asymptotic Expansion of n!, J. Int. Seq. 13 (2010), 10.6.6.

FORMULA

S_r(n+1,k) = k S_r(n,k) + binomial(n,r-1)*S_r(n-r+1,k-1); for this sequence, r=3.

G.f.: Sum_{n>=0, k>=0} S_r(n,k)*u^k*t^n/n! = exp(u(e^t - Sum_{i=0..r-1} t^i/i!)).

EXAMPLE

There are 10 ways of partitioning a set N of cardinality 6 into 2 blocks each of cardinality at least 3, so S_3(6,2) = 10.

MATHEMATICA

S3[3, 1] = S3[4, 1] = S3[5, 1] = 1; S3[n_, k_] /; 1 <= k <= Floor[n/3] := S3[n, k] = k*S3[n-1, k] + Binomial[n-1, 2]*S3[n-3, k-1]; S3[_, _] = 0; Flatten[ Table[ S3[n, k], {n, 3, 20}, {k, 1, Floor[n/3]}]] (* Jean-Fran├žois Alcover, Feb 21 2012 *)

CROSSREFS

Cf. A008299, A059023, A059024, A059025.

Sequence in context: A070246 A085044 A215268 * A193634 A115097 A050313

Adjacent sequences:  A059019 A059020 A059021 * A059023 A059024 A059025

KEYWORD

nonn,tabf,nice

AUTHOR

Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 14 2000

STATUS

approved

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Last modified August 23 04:06 EDT 2017. Contains 290958 sequences.