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Search: id:A124422
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| A124422 |
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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 only even entries (0<=k<=floor(n/2)). |
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8
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| 1, 1, 1, 1, 3, 2, 5, 8, 2, 22, 25, 5, 52, 101, 45, 5, 283, 423, 156, 15, 855, 1889, 1143, 238, 15, 5451, 9726, 5002, 916, 52, 19921, 48382, 35805, 10540, 1275, 52, 144074, 292223, 187515, 49155, 5400, 203, 614866, 1609551, 1379753, 512710, 89425, 7089, 203
(list; graph; listen)
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OFFSET
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0,5
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COMMENT
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Row n has 1+floor(n/2) terms. Row sums are the Bell numbers (A000110). T(2n-1,n-1)=T(2n,n)=A000110(n) (the Bell numbers). T(n,0)=A124423(n).
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FORMULA
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The generating polynomial of row n is P[n](s)=Q[n](1,s,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
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T(4,1)=8 because we have 134|2, 13|24, 14|2|3, 1|24|3, 1|2|34, 123|4, 1|23|4, and 12|3|4.
Triangle starts:
1;
1;
1,1;
3,2;
5,8,2;
22,25,5;
52,101,45,5;
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MAPLE
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Q[0]:=1: for n from 1 to 13 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 13 do P[n]:=sort(subs({t=1, x=1}, Q[n])) od: for n from 0 to 13 do seq(coeff(P[n], s, j), j=0..floor(n/2)) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000110, A124418, A124419, A124420, A124421, A124423.
Sequence in context: A094894 A089334 A016649 this_sequence A132776 A019594 A085167
Adjacent sequences: A124419 A124420 A124421 this_sequence A124423 A124424 A124425
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 31 2006
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