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A059022
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Triangle of Stirling numbers of order 3.
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12
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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, 297797500
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OFFSET
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3,5
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COMMENTS
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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. The initial entry in the sequence is S_3(3,1).
Rows are of lengths 1,1,1,2,2,2,3,3,3,...
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.
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LINKS
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Table of n, a(n) for n=3..46.
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.
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FORMULA
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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!)).
T(n,k) = Sum_{j=0..min(n/2,k)} (-1)^j*B(n,j)*S_2(n-2j,k-j), where B are the Bessel numbers A100861 and S_2 are the 2-associated Stirling numbers of the second kind A008299. - Fabián Pereyra, Feb 20 2022
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EXAMPLE
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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.
From Wesley Ivan Hurt, Feb 24 2022: (Start)
Triangle starts:
1;
1;
1;
1, 10;
1, 35;
1, 91;
1, 210, 280;
1, 456, 2100;
1, 957, 10395;
1, 1969, 42735, 15400;
...
(End)
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MAPLE
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b:= proc(n) option remember; `if`(n=0, 1, add(
expand(x*b(n-j))*binomial(n-1, j-1), j=3..n))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
seq(T(n), n=3..20); # Alois P. Heinz, Feb 21 2022
# alternative
A059022 := proc(n, k)
option remember;
if n<3 then
0;
elif n < 6 and k=1 then
1 ;
else
k*procname(n-1, k)+binomial(n-1, 2)*procname(n-3, k-1) ;
end if;
end proc: # R. J. Mathar, Apr 15 2022
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MATHEMATICA
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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-François Alcover, Feb 21 2012 *)
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CROSSREFS
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Row sums give A006505.
Cf. A008299, A059023, A059024, A059025, A100861, A272352 (column 2), A272982 (column 3), A261724 (column 4), A352611 (column 5).
Sequence in context: A070246 A085044 A215268 * A193634 A115097 A050313
Adjacent sequences: A059019 A059020 A059021 * A059023 A059024 A059025
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KEYWORD
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nonn,tabf,nice
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AUTHOR
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Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 14 2000
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STATUS
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approved
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