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A213744 Triangle of numbers C^(5)(n,k) of combinations with repetitions from n different elements over k for each of them not more than 5 appearances allowed.

%I A213744 #18 Apr 24 2013 12:24:11
%S A213744 1,1,1,1,2,3,1,3,6,10,1,4,10,20,35,1,5,15,35,70,126,1,6,21,56,126,252,
%T A213744 456,1,7,28,84,210,462,917,1667,1,8,36,120,330,792,1708,3368,6147,1,9,
%U A213744 45,165,495,1287,2994,6354,12465,22825,1,10
%N A213744 Triangle of numbers C^(5)(n,k) of combinations with repetitions from n different elements over k for each of them not more than 5 appearances allowed.
%C A213744 For k<=4, the triangle coincides with triangle A213743.
%C A213744 We have over columns of the triangle: 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)=A000332(n) for n>=7, T(n,5)=A000389(n) for n>=9, T(n,6)=A062989(n) for n>=5, T(n,7)=A063262 for n>=5, T(n,8)=A063263 for n>=6, T(n,9)=A063264 for n>=7.
%H A213744 Peter J. C. Moses, <a href="/A213744/b213744.txt">Rows n = 0..50 of triangle, flattened</a>
%F A213744 C^(5)(n,k)=sum{r=0,...,floor(k/6)}(-1)^r*C(n,r)*C(n-6*r+k-1, n-1)
%e A213744 Triangle begins
%e A213744 n/k.|..0.....1.....2.....3.....4.....5.....6.....7
%e A213744 ==================================================
%e A213744 .0..|..1
%e A213744 .1..|..1.....1
%e A213744 .2..|..1.....2.....3
%e A213744 .3..|..1.....3.....6....10
%e A213744 .4..|..1.....4....10....20....35
%e A213744 .5..|..1.....5....15....35....70....126
%e A213744 .6..|..1.....6....21....56...126....252...456
%e A213744 .7..|..1.....7....28....84...210....462...917....1667
%t A213744 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}]&[6] (* _Peter J. C. Moses_, Apr 16 2013 *)
%Y A213744 Cf. A007318, A005725, A111808, A187925, A213742, A213743, A000217, A000292, A000332, A000389, A062989, A063262, A063263, A063264.
%K A213744 nonn,tabl
%O A213744 0,5
%A A213744 _Vladimir Shevelev_ and _Peter J. C. Moses_, Jun 19 2012