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A213742
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Triangle of numbers C^(3)(n,k) of combinations with repetitions from n different elements over k for each of them not more than three appearances allowed.
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7
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1, 1, 1, 1, 2, 3, 1, 3, 6, 10, 1, 4, 10, 20, 31, 1, 5, 15, 35, 65, 101, 1, 6, 21, 56, 120, 216, 336, 1, 7, 28, 84, 203, 413, 728, 1128, 1, 8, 36, 120, 322, 728, 1428, 2472, 3823, 1, 9, 45, 165, 486, 1206, 2598, 4950, 8451, 13051, 1, 10
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OFFSET
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0,5
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COMMENTS
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The left side of triangle consists of 1's, while the right side is formed by A005725. Further, T(n,0)=1, T(n,1)=n, T(n,2)=A000217(n) for n>1, T(n,3)=A000292(n) for n>=3, T(n,4)=A005718(n) for n>=2, T(n,5)=A005719(n) for n>=5, T(n,6)=A005720(n) for n>=6, T(n,7)=A001919(n) for n>=7, T(n,8)=A064055(n) for n>=5.
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LINKS
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FORMULA
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C^(3)(n,k)=sum{r=0,...,floor(k/4)}(-1)^r*C(n,r)*C(n-4*r+k-1, n-1)
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EXAMPLE
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Triangle begins
n/k.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1.....1
.2..|..1.....2.....3
.3..|..1.....3.....6....10
.4..|..1.....4....10....20....31
.5..|..1.....5....15....35....65....101
.6..|..1.....6....21....56...120....216...336
.7..|..1.....7....28....84...203....413...728....1128
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MATHEMATICA
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Flatten[Table[Sum[(-1)^r Binomial[n, r] Binomial[n-# r+k-1, n-1], {r, 0, Floor[k/#]}], {n, 0, 15}, {k, 0, n}]/.{0}->{1}]&[4] (* Peter J. C. Moses, Apr 16 2013 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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