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A059024 Triangle of Stirling numbers of order 5. 5
1, 1, 1, 1, 1, 1, 126, 1, 462, 1, 1254, 1, 3003, 1, 6721, 1, 14443, 126126, 1, 30251, 1009008, 1, 62322, 5309304, 1, 127024, 23075052, 1, 257108, 89791416, 1, 518092, 325355316, 488864376, 1, 1041029, 1122632043, 6844101264, 1, 2088043 (list; graph; refs; listen; history; text; internal format)
OFFSET

5,7

COMMENTS

The number of partitions of the set N, |N|=n, into k blocks, all of cardinality greater than or equal to 5. This is the 5-associated Stirling number of the second kind.

This is entered as a triangular array. The entries S_5(n,k) are zero for 5k>n, so these values are omitted. Initial entry in sequence is S_5(5,1).

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

REFERENCES

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

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

LINKS

Alois P. Heinz, Rows n = 5..320, flattened

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

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=5.

G.f.: sum(S_r(n, k)u^k ((t^n)/(n!)), n=0..infty, k=0..infty) = exp(u(e^t-sum(t^i/i!, i=0..r-1))).

EXAMPLE

There are 126 ways of partitioning a set N of cardinality 10 into 2 blocks each of cardinality at least 5, so S_5(10,2) = 126.

Triangle begins:

1;

1;

1;

1;

1;

1,    126;

1,    462;

1,   1254;

1,   3003;

1,   6721;

1,  14443,    126126;

1,  30251,   1009008;

1,  62322,   5309304;

1, 127024,  23075052;

1, 257108,  89791416;

1, 518092, 325355316, 488864376;

...

MAPLE

T:= proc(n, k) option remember; `if`(k<1 or k>n/5, 0,

      `if`(k=1, 1, k*T(n-1, k)+binomial(n-1, 4)*T(n-5, k-1)))

    end:

seq(seq(T(n, k), k=1..n/5), n=5..25);  # Alois P. Heinz, Aug 18 2017

MATHEMATICA

S5[n_ /; 5 <= n <= 9, 1] = 1; S5[n_, k_] /; 1 <= k <= Floor[n/5] := S5[n, k] = k*S5[n-1, k] + Binomial[n-1, 4]*S5[n-5, k-1]; S5[_, _] = 0; Flatten[ Table[ S5[n, k], {n, 5, 25}, {k, 1, Floor[n/5]}]] (* Jean-Fran├žois Alcover, Feb 21 2012 *)

CROSSREFS

Cf. A008299, A059022, A059023, A059025.

Sequence in context: A318257 A110825 A050451 * A267199 A267403 A267342

Adjacent sequences:  A059021 A059022 A059023 * A059025 A059026 A059027

KEYWORD

nonn,tabf,nice

AUTHOR

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

STATUS

approved

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Last modified October 17 16:51 EDT 2019. Contains 328120 sequences. (Running on oeis4.)