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 A124503 Triangle read by rows: T(n,k) is the number of set partitions of the set {1,2,...,n} (or of any n-set) containing k blocks of size 3 (0<=k<=floor(n/3)). 4
 1, 1, 2, 4, 1, 11, 4, 32, 20, 113, 80, 10, 422, 385, 70, 1788, 1792, 560, 8015, 9492, 3360, 280, 39435, 50640, 23100, 2800, 204910, 295020, 147840, 30800, 1144377, 1763300, 1044120, 246400, 15400, 6722107, 11278410, 7241520, 2202200, 200200, 41877722 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Row n contains 1+floor(n/3) terms. Row sums yield the Bell numbers (A000110). T(n,0)=A124504(n). Sum(k*T(n,k), k=0..floor(n/3))=A105480(n+1). LINKS Alois P. Heinz, Rows n = 0..250, flattened FORMULA E.g.f.: G(t,z) = exp(exp(z)-1+(t-1)z^3/6). EXAMPLE T(4,1)=4 because we have 1|234, 134|2, 124|3 and 123|4. Triangle starts: 1; 1; 2; 4, 1; 11, 4; 32, 20; 113, 80, 10; 422, 385, 70; ... MAPLE G:=exp(exp(z)-1+(t-1)*z^3/6): Gser:=simplify(series(G, z=0, 17)): for n from 0 to 14 do P[n]:=sort(n!*coeff(Gser, z, n)) od: for n from 0 to 14 do seq(coeff(P[n], t, k), k=0..floor(n/3)) 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`(i=3, 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 = 8; k = 3; Range[0, nn]! CoefficientList[ Series[Exp[Exp[x] - 1 + (y - 1) x^k/k!], {x, 0, nn}], {x, y}] // Grid // Geoffrey Critzer, Aug 26 2012 CROSSREFS Cf. A000110, A124504, A105480, A355144. T(3n,n) gives A025035. Sequence in context: A308300 A246188 A135333 * A114499 A030730 A117131 Adjacent sequences: A124500 A124501 A124502 * A124504 A124505 A124506 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Nov 14 2006 STATUS approved

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Last modified December 7 07:00 EST 2022. Contains 358649 sequences. (Running on oeis4.)