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A213808 Triangle of numbers C^(7)(n,k) of combinations with repetitions from n different elements over k for each of them not more than 7 appearances allowed.

%I A213808 #20 Nov 26 2017 09:48:30
%S A213808 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 A213808 462,1,7,28,84,210,462,924,1716,1,8,36,120,330,792,1716,3432,6427,1,9,
%U A213808 45,165,495,1287,3003,6435,12861,24229,1,10,55,220,715,2002,5005,11440,24300,48520,91828
%N A213808 Triangle of numbers C^(7)(n,k) of combinations with repetitions from n different elements over k for each of them not more than 7 appearances allowed.
%C A213808 For k <= 6, the triangle coincides with triangle A213745.
%H A213808 G. C. Greubel, <a href="/A213808/b213808.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A213808 T(n,k) = Sum_{r=0..floor(k/8)} (-1)^r*C(n,r)*C(n-8*r+k-1, n-1).
%F A213808 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)=A000579(n) for n >= 11, T(n,7)=A000580(n) for n >= 13.
%e A213808 Triangle begins
%e A213808 n/k |  0     1     2     3     4     5     6     7     8
%e A213808 ----+---------------------------------------------------
%e A213808   0 |  1
%e A213808   1 |  1     1
%e A213808   2 |  1     2     3
%e A213808   3 |  1     3     6    10
%e A213808   4 |  1     4    10    20    35
%e A213808   5 |  1     5    15    35    70   126
%e A213808   6 |  1     6    21    56   126   252   462
%e A213808   7 |  1     7    28    84   210   462   924  1716
%e A213808   8 |  1     8    36   120   330   792  1716  3432  6427
%t A213808 Table[Sum[(-1)^r*Binomial[n, r]*Binomial[n - 8*r + k - 1, n - 1], {r, 0, Floor[k/8]}], {n, 0, 10}, {k, 0, n}] // Flatten (* _G. C. Greubel_, Nov 25 2017 *)
%o A213808 (PARI) for(n=0,10, for(k=0,n, print1(if(n==0 && k==0, 1, sum(r=0, floor(k/8), (-1)^r*binomial(n,r)*binomial(n-8*r + k-1,n-1))), ", "))) \\ _G. C. Greubel_, Nov 25 2017
%Y A213808 Cf. A007318, A005725, A059481, A111808, A187925, A213742, A213743, A213744, A000217, A000292, A000332, A000389, A000579, A000580.
%K A213808 nonn,tabl
%O A213808 0,5
%A A213808 _Vladimir Shevelev_ and _Peter J. C. Moses_, Jun 20 2012