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 A059024 Triangle of Stirling numbers of order 5. 6
 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_{n>=0, k>=0} S_r(n,k)*u^k*t^n/n! = exp(u(e^t-sum(t^i/i!, i=0..r-1))). T(n,k) = Sum_{j=0..min(n/4,k)} (-1)^j*n!/(24^j*j!*(n-4j)!)*S_4(n-4j,k-j), where S_4 are the 4-associated Stirling numbers of the second kind A059023. - Fabián Pereyra, Feb 21 2022 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-François Alcover, Feb 21 2012 *) CROSSREFS Row sums give A057814. Cf. A008299, A059022, A059023, A059025. Sequence in context: A110825 A365897 A050451 * A365896 A331401 A267199 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 July 20 16:05 EDT 2024. Contains 374459 sequences. (Running on oeis4.)