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A124498
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)).
2
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
OFFSET
0,5
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).
LINKS
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
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.
Triangle T(n,k) begins:
: 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, 62370, 10395;
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
# Alternative:
with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
`if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
b(n-i*j, i-1)*`if`(i=2, x^j, 1), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..15); # Alois P. Heinz, Mar 08 2015
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 *)
CROSSREFS
T(2n,n) gives A001147.
Sequence in context: A102402 A246129 A359123 * A197334 A113399 A085273
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Nov 05 2006
STATUS
approved