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A197126
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Triangle T(n,k), n>=1, 1<=k<=n, read by rows: T(n,k) is the number of cliques of size k in all partitions of n.
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11
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1, 1, 1, 3, 0, 1, 4, 2, 0, 1, 8, 2, 1, 0, 1, 11, 4, 2, 1, 0, 1, 19, 5, 3, 1, 1, 0, 1, 26, 10, 3, 3, 1, 1, 0, 1, 41, 11, 7, 3, 2, 1, 1, 0, 1, 56, 20, 8, 5, 3, 2, 1, 1, 0, 1, 83, 25, 13, 6, 5, 2, 2, 1, 1, 0, 1, 112, 38, 17, 11, 5, 5, 2, 2, 1, 1, 0, 1, 160, 49, 25, 13, 9, 5, 4, 2, 2, 1, 1, 0, 1
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OFFSET
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1,4
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COMMENTS
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All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.
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LINKS
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Alois P. Heinz, Rows n = 1..141, flattened
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FORMULA
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G.f. of column k: (x^k/(1-x^k)-x^(k+1)/(1-x^(k+1)))/Product_{j>0}(1-x^j).
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EXAMPLE
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T(4,1) = 4: [1,1,(2)], [(1),(3)], [(4)].
T(8,3) = 3: [1,1,(2,2,2)], [(1,1,1),2,3], [(1,1,1),5].
T(12,4) = 11: [(1,1,1,1),(2,2,2,2)], [1,(2,2,2,2),3], [(1,1,1,1),2,3,3], [(3,3,3,3)], [(1,1,1,1),2,2,4], [(2,2,2,2),4], [(1,1,1,1),4,4], [(1,1,1,1),3,5], [(1,1,1,1),2,6], [(1,1,1,1),8]. Here the first partition contains 2 cliques.
Triangle begins:
1;
1, 1;
3, 0, 1;
4, 2, 0, 1;
8, 2, 1, 0, 1;
11, 4, 2, 1, 0, 1;
19, 5, 3, 1, 1, 0, 1;
26, 10, 3, 3, 1, 1, 0, 1;
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MAPLE
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b:= proc(n, p, k) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
add((l->`if`(m=k, l+[0, l[1]], l))(b(n-p*m, p-1, k)), m=0..n/p)))
end:
T:= (n, k)-> b(n, n, k)[2]:
seq(seq(T(n, k), k=1..n), n=1..20);
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MATHEMATICA
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Table[CoefficientList[ 1/q* Tr[Flatten[q^Map[Length, Split /@ Partitions[n], {2}]]], q], {n, 24}] (* Wouter Meeussen, Apr 21 2012 *)
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CROSSREFS
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Columns k=1-10 give: A024786(n+1), A116646, A117524, A222704, A222705, A222706, A222707, A222708, A222709, A222710.
Row sums give: A000070(n-1). Diagonal gives: A000012. Limit of reversed rows: T(2*n+1,n+1) = A002865(n).
Cf. A213180.
Sequence in context: A115142 A178116 A177330 * A048963 A119458 A106356
Adjacent sequences: A197123 A197124 A197125 * A197127 A197128 A197129
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KEYWORD
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nonn,tabl
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AUTHOR
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Alois P. Heinz, Oct 10 2011
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STATUS
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approved
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