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A124496
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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.
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2
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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FORMULA
| The row enumerating polynomial P[n](t)=Q[n](t,1), where Q[1](t,s)=ts and Q[n](t,s)=s*dQ[n-1](t,s)/ds +(t-1)Q[n-1](t,s)+tsQ[n-1](1,s) for n>=2.
A008275^-1*ONES*A008275 or A008277*ONES*A008277^-1 where ONES is a triangle with all entries = 1. [From Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Aug 20 2009]
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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;
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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;
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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.
Cf. A000110, A040027, A045501, A045499, A045500.
Sequence in context: A100537 A069605 A080510 * A074881 A142992 A145905
Adjacent sequences: A124493 A124494 A124495 * A124497 A124498 A124499
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 14 2006
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