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A124498
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Triangle read by rows: T(n,k) is the number of set partitions of the set {1,2,...,n} containing k blocks of size 2 (0 <= k <= floor(n/2)).
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1
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1, 1, 1, 1, 2, 3, 6, 6, 3, 17, 20, 15, 53, 90, 45, 15, 205, 357, 210, 105, 871, 1484, 1260, 420, 105, 3876, 7380, 6426, 2520, 945, 18820, 39195, 33390, 18900, 4725, 945, 99585, 213180, 202950, 117810, 34650, 10395, 558847, 1242120, 1293435, 734580, 311850
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Row n contains 1+floor(n/2) terms. Row sums yield the Bell numbers A000110. T(n,0)=A097514(n). Sum(k*T(n,k), k=0..floor(n/2))=A105479(n+1).
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FORMULA
| E.g.f.=exp(exp(z)-1+(t-1)z^2/2).
Generally the e.g.f. for set partitions containing k blocks of size p is: G(z,t) = exp(exp(z)-1+(t-1)z^p/p!) - Geoffrey Critzer, Nov 30 2011
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EXAMPLE
| T(4,1)=6 because we have 12|3|4, 13|2|4, 14|2|3, 1|23|4, 1|24|3 and 1|2|34.
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MAPLE
| G:=exp(exp(z)-1+(t-1)*z^2/2): Gser:=simplify(series(G, z=0, 16)): for n from 0 to 13 do P[n]:=sort(n!*coeff(Gser, z, n)) od: for n from 0 to 13 do seq(coeff(P[n], t, k), k=0..floor(n/2)) od; # yields sequence in triangular form
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MATHEMATICA
| d=Exp[Exp[x]-x^2/2!-1]; f[list_] := Select[list, #>0&]; Map[f, Transpose[Table[Range[0, 12]! CoefficientList[Series[ x^(2k)/(k! 2!^k) *d, {x, 0, 12}], x], {k, 0, 5}]]]//Flatten (* Geoffrey Critzer, Nov 30 2011 *)
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CROSSREFS
| Cf. A000110, A097514, A105479, A124503.
Sequence in context: A066838 A084678 A102402 * A197334 A113399 A085273
Adjacent sequences: A124495 A124496 A124497 * A124499 A124500 A124501
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 05 2006
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