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A124321
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Triangle read by rows: T(n,k) is the number of set partitions of {1,2,...,n} (or of any n-set) having k blocks of odd size (0<=k<=n).
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1
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1, 0, 1, 1, 0, 1, 0, 4, 0, 1, 4, 0, 10, 0, 1, 0, 31, 0, 20, 0, 1, 31, 0, 136, 0, 35, 0, 1, 0, 379, 0, 441, 0, 56, 0, 1, 379, 0, 2500, 0, 1176, 0, 84, 0, 1, 0, 6556, 0, 11740, 0, 2730, 0, 120, 0, 1, 6556, 0, 59671, 0, 43870, 0, 5712, 0, 165, 0, 1, 0, 150349, 0, 378356, 0, 138622, 0
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,8
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COMMENTS
| Row sums are the Bell numbers (A000110). Sum(k*T(n,k),k=0..n)=A102286(n). T(2n,0)=A005046(n); T(2n+1,0)=0.
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 225.
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FORMULA
| E.g.f.=G(t,z)=exp[t*sinh(z)+cosh(z)-1].
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EXAMPLE
| T(3,1)=4 because we have 123, 1|23, 12|3 and 13|2.
Triangle starts:
1;
0,1;
1,0,1;
0,4,0,1;
4,0,10,0,1;
0,31,0,20,0,1;
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MAPLE
| G:=exp(t*sinh(z)+cosh(z)-1): Gser:=simplify(series(G, z=0, 15)): for n from 0 to 12 do P[n]:=sort(n!*coeff(Gser, z, n)) od: for n from 0 to 12 do seq(coeff(P[n], t, j), j=0..n) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A000110, A102286, A005046, A124322.
Sequence in context: A115636 A178104 A172545 * A100045 A143844 A186759
Adjacent sequences: A124318 A124319 A124320 * A124322 A124323 A124324
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 28 2006
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