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A213743 Triangle T(n,k), read by rows, of numbers T(n,k)=C^(4)(n,k) of combinations with repetitions from n different elements over k for each of them not more than four appearances allowed.

%I A213743 #27 Apr 21 2025 13:22:54
%S A213743 1,1,1,1,2,3,1,3,6,10,1,4,10,20,35,1,5,15,35,70,121,1,6,21,56,126,246,
%T A213743 426,1,7,28,84,210,455,875,1520,1,8,36,120,330,784,1652,3144,5475,1,9,
%U A213743 45,165,495,1278,2922,6030,11385,19855,1,10,55,220,715,1992,4905,10890,22110,41470,72403
%N A213743 Triangle T(n,k), read by rows, of numbers T(n,k)=C^(4)(n,k) of combinations with repetitions from n different elements over k for each of them not more than four appearances allowed.
%C A213743 The left side of triangle consists of 1's, while the right side is formed by A187925. 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)=A000332(n) for n>=7, T(n,5)=A027659(n) for n>=3, T(n,6)=A064056(n) for n>=4, T(n,7)=A064057(n) for n>=5, T(n,8)=A064058(n) for n>=6, T(n,9)=A000575(n) for n>=6.
%H A213743 Peter J. C. Moses, <a href="/A213743/b213743.txt">Rows n = 0..50 of triangle, flattened</a>
%F A213743 C^(4)(n,k) = Sum_{r=0...floor(k/5)} (-1)^r*C(n,r)*C(n-5*r+k-1, n-1).
%e A213743 Triangle begins
%e A213743   n/k.|..0.....1.....2.....3.....4.....5.....6.....7
%e A213743   ==================================================
%e A213743   .0..|..1
%e A213743   .1..|..1.....1
%e A213743   .2..|..1.....2.....3
%e A213743   .3..|..1.....3.....6....10
%e A213743   .4..|..1.....4....10....20....35
%e A213743   .5..|..1.....5....15....35....70....121
%e A213743   .6..|..1.....6....21....56...126....246...426
%e A213743   .7..|..1.....7....28....84...210....455...875....1520
%e A213743 T(4,2)=C^(4)(4,2): From 4 elements {1,2,3,4}, we have the following 10 allowed combinations of 2 elements: {1,1}, {1,2}, {1,3}, {1,4}, {2,2}, {2,3}, {2,4}, {3,3}, {3,4}, {4,4}.
%t A213743 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}]&[5] (* _Peter J. C. Moses_, Apr 16 2013 *)
%Y A213743 Cf. A007318, A005725, A111808, A187925, A213742, A000217, A000292, A000332, A027659, A064056, A064057, A064058, A000575.
%K A213743 nonn,tabl
%O A213743 0,5
%A A213743 _Vladimir Shevelev_ and _Peter J. C. Moses_, Jun 19 2012