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A124424
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Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n}, having exactly k blocks consisting of entries of the same parity (0<=k<=n).
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1
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1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 3, 4, 5, 2, 1, 7, 14, 16, 10, 4, 1, 25, 48, 61, 42, 20, 6, 1, 79, 194, 250, 200, 106, 38, 9, 1, 339, 820, 1145, 958, 569, 230, 66, 12, 1, 1351, 3794, 5554, 5096, 3251, 1486, 486, 112, 16, 1, 6721, 18960, 29101, 28010, 19110, 9470, 3477, 930
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,8
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COMMENTS
| Row sums are the Bell numbers (A000110). T(n,0)=A124425(n).
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FORMULA
| The generating polynomial of row n is P[n](t)=Q[n](t,t,1), where the polynomials Q[n]=Q[n](t,s,x) are defined by Q[0]=1; Q[n]=t*dQ[n-1]/dt + x*dQ[n-1]/ds + x*dQ[n-1]/dx + t*Q[n-1] if n is odd and Q[n]=x*dQ[n-1]/dt + s*dQ[n-1]/ds + x*dQ[n-1]/dx + s*Q[n-1] if n is even.
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EXAMPLE
| T(4,2)=5 because we have 13|24, 14|2|3, 1|2|34, 1|23|4 and 12|3|4.
Triangle starts:
1;
0,1;
1,0,1;
1,2,1,1;
3,4,5,2,1;
7,14,16,10,4,1;
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MAPLE
| Q[0]:=1: for n from 1 to 11 do if n mod 2 = 1 then Q[n]:=expand(t*diff(Q[n-1], t)+x*diff(Q[n-1], s)+x*diff(Q[n-1], x)+t*Q[n-1]) else Q[n]:=expand(x*diff(Q[n-1], t)+s*diff(Q[n-1], s)+x*diff(Q[n-1], x)+s*Q[n-1]) fi od: for n from 0 to 11 do P[n]:=sort(subs({s=t, x=1}, Q[n])) od: for n from 0 to 11 do seq(coeff(P[n], t, j), j=0..n) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A000110, A124418, A124419, A124420, A124421, A124422, A124423, A124425.
Sequence in context: A015138 A157807 A100529 * A057044 A153899 A068098
Adjacent sequences: A124421 A124422 A124423 * A124425 A124426 A124427
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 01 2006
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