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A124322 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 even size (0<=k<=floor(n/2)). 3
1, 1, 1, 1, 2, 3, 5, 7, 3, 12, 25, 15, 37, 91, 60, 15, 128, 329, 315, 105, 457, 1415, 1533, 630, 105, 1872, 6297, 7623, 4410, 945, 8169, 29431, 42150, 27405, 7875, 945, 37600, 151085, 233475, 176715, 69300, 10395, 188685, 802099, 1365243, 1199220, 533610 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row n has 1+floor(n/2) terms. Sum of row n is the Bell number B(n)=A000110(n). Sum(k*T(n,k),k=0..floor(n/2))=A102287(n). T(n,0)=A003724(n).

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 225.

LINKS

Alois P. Heinz, Rows n = 0..200, flattened

FORMULA

E.g.f.: exp[sinh(z)+t(cosh(z)-1)].

EXAMPLE

T(4,1) = 7 because we have 1234, 14|2|3, 1|24|3, 1|2|34, 13|2|4, 1|23|4 and 12|3|4.

Triangle starts:

1;

1;

1,1;

2,3;

5,7,3;

12,25,15;

37,91,60,15;

MAPLE

G:=exp(sinh(z)+t*(cosh(z)-1)): 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, j), j=0..floor(n/2)) od; # yields sequence in triangular form

# second Maple program:

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`(irem(i, 2)=0, 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

nn = 10; Range[0, nn]! CoefficientList[Series[Exp[y (Cosh[x] - 1) + Sinh[x]], {x, 0, nn}], {x, y}] // Grid  (* Geoffrey Critzer, Aug 28 2012*)

CROSSREFS

Cf. A000110, A102887, A003724, A124321.

Sequence in context: A163821 A156294 A126607 * A209037 A334455 A075241

Adjacent sequences:  A124319 A124320 A124321 * A124323 A124324 A124325

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Oct 28 2006

STATUS

approved

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Last modified September 22 19:36 EDT 2021. Contains 347608 sequences. (Running on oeis4.)