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Search: id:A124420
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| A124420 |
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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 odd entries (0<=k<=ceil(n/2)). |
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8
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| 1, 0, 1, 1, 1, 1, 3, 1, 5, 8, 2, 9, 26, 15, 2, 52, 101, 45, 5, 130, 385, 287, 70, 5, 855, 1889, 1143, 238, 15, 2707, 8295, 7320, 2475, 335, 15, 19921, 48382, 35805, 10540, 1275, 52, 75771, 240534, 240082, 100940, 19505, 1686, 52, 614866, 1609551, 1379753, 512710
(list; graph; listen)
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OFFSET
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0,7
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COMMENT
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Row n has 1+ceil(n/2) terms. Row sums are the Bell numbers (A000110). T(2n,n)=T(2n+1,n+1)=A000110(n) (the Bell numbers). T(n,0)=A124421(n).
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FORMULA
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The generating polynomial of row n is P[n](t)=Q[n](t,1,1), 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 13|24, 1|234, 124|3, 14|2|3, 1|2|34, 13|2|4, 1|23|4, and 12|3|4.
Triangle starts:
1;
0,1;
1,1;
1,3,1;
5,8,2;
9,26,15,2;
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({s=1, x=1}, Q[n])) od: for n from 0 to 13 do seq(coeff(P[n], t, j), j=0..ceil(n/2)) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000110, A124418, A124419, A124421, A124422, A124423.
Adjacent sequences: A124417 A124418 A124419 this_sequence A124421 A124422 A124423
Sequence in context: A038738 A116647 A063858 this_sequence A094353 A129801 A128821
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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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