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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.
2

%I #20 Nov 26 2017 09:48:30

%S 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 462,1,7,28,84,210,462,924,1716,1,8,36,120,330,792,1716,3432,6427,1,9,

%U 45,165,495,1287,3003,6435,12861,24229,1,10,55,220,715,2002,5005,11440,24300,48520,91828

%N 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 For k <= 6, the triangle coincides with triangle A213745.

%H G. C. Greubel, <a href="/A213808/b213808.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%F T(n,k) = Sum_{r=0..floor(k/8)} (-1)^r*C(n,r)*C(n-8*r+k-1, n-1).

%F 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 Triangle begins

%e n/k | 0 1 2 3 4 5 6 7 8

%e ----+---------------------------------------------------

%e 0 | 1

%e 1 | 1 1

%e 2 | 1 2 3

%e 3 | 1 3 6 10

%e 4 | 1 4 10 20 35

%e 5 | 1 5 15 35 70 126

%e 6 | 1 6 21 56 126 252 462

%e 7 | 1 7 28 84 210 462 924 1716

%e 8 | 1 8 36 120 330 792 1716 3432 6427

%t 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 (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 Cf. A007318, A005725, A059481, A111808, A187925, A213742, A213743, A213744, A000217, A000292, A000332, A000389, A000579, A000580.

%K nonn,tabl

%O 0,5

%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Jun 20 2012