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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 (list; graph; refs; listen; history; text; internal format)
 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 Alois P. Heinz, Rows n = 0..200, flattened 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 # 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=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 Cf. A000110, A097514, A105479, A124503. T(2n,n) gives A001147. Sequence in context: A102402 A246129 A359123 * A197334 A113399 A085273 Adjacent sequences: A124495 A124496 A124497 * A124499 A124500 A124501 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Nov 05 2006 STATUS approved

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Last modified April 18 22:18 EDT 2024. Contains 371782 sequences. (Running on oeis4.)