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 A124496 Triangle read by rows: T(n,k) is the number of set partitions of {1,2,...,n} in which the size of the last block is k, 1<=k<=n; the blocks are ordered with increasing least elements. 8
 1, 1, 1, 3, 1, 1, 9, 4, 1, 1, 31, 14, 5, 1, 1, 121, 54, 20, 6, 1, 1, 523, 233, 85, 27, 7, 1, 1, 2469, 1101, 400, 125, 35, 8, 1, 1, 12611, 5625, 2046, 635, 175, 44, 9, 1, 1, 69161, 30846, 11226, 3488, 952, 236, 54, 10, 1, 1, 404663, 180474, 65676, 20425, 5579, 1366, 309, 65, 11, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Number of restricted growth functions of length n with a multiplicity k of the maximum value. RGF's are here defined as f(1)=1, f(i) <= 1+max_{1<=j=2. A008275^-1*ONES*A008275 or A008277*ONES*A008277^-1 where ONES is a triangle with all entries = 1. [From Gerald McGarvey, Aug 20 2009] Conjectures: T(n,n-3) = A000096(n). T(n,n-4)= A055831(n+1). - R. J. Mathar, Mar 13 2016 EXAMPLE T(4,2)=4 because we have 13|24, 14|23, 12|34 and 1|2|34. Triangle starts: 1; 1,1; 3,1,1; 9,4,1,1; 31,14,5,1,1; 121,54,20,6,1,1; 523,233,85,27,7,1,1; 2469,1101,400,125,35,8,1,1; 12611,5625,2046,635,175,44,9,1,1; 69161,30846,11226,3488,952,236,54,10,1,1; 404663,180474,65676,20425,5579,1366,309,65,11,1,1; 2512769,1120666,407787,126817,34685,8494,1893,395,77,12,1,1; MAPLE Q[1]:=t*s: for n from 2 to 12 do Q[n]:=expand(t*s*subs(t=1, Q[n-1])+s*diff(Q[n-1], s)+t*Q[n-1]-Q[n-1]) od:for n from 1 to 12 do P[n]:=sort(subs(s=1, Q[n])) od: for n from 1 to 12 do seq(coeff(P[n], t, j), j=1..n) od; # second Maple program: T:= proc(n, k) option remember; `if`(n=k, 1,       add(T(n-j, k)*binomial(n-1, j-1), j=1..n-k))     end: seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Jul 05 2016 MATHEMATICA T[n_, k_] := T[n, k] = If[n == k, 1, Sum[T[n-j, k]*Binomial[n-1, j-1], {j, 1, n-k}]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten; (* Jean-François Alcover, Jul 21 2016, after Alois P. Heinz *) CROSSREFS Row sums are the Bell numbers (A000110). It seems that T(n, 1), T(n, 2), T(n, 3) and T(n, 4) are given by A040027, A045501, A045499 and A045500, respectively. A121207 gives a very similar triangle. T(2n,n) gives A297924. Cf. A000110, A040027, A045501, A045499, A045500. Sequence in context: A100537 A069605 A080510 * A074881 A142992 A145905 Adjacent sequences:  A124493 A124494 A124495 * A124497 A124498 A124499 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Nov 14 2006 STATUS approved

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Last modified January 17 10:59 EST 2019. Contains 319218 sequences. (Running on oeis4.)