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A259074 Triangle T(n,k) = Sum_{j=0..(n-k)/3} C(n-3*j-1,k-1)*C(n-k-3*j,j).

%I A259074 #16 Mar 19 2024 08:19:08
%S A259074 1,1,1,1,2,1,1,3,3,1,2,4,6,4,1,3,7,10,10,5,1,4,12,18,20,15,6,1,5,19,
%T A259074 33,39,35,21,7,1,7,28,58,76,75,56,28,8,1,10,42,96,144,156,132,84,36,9,
%U A259074 1,14,64,156,260,315,294,217,120,45,10,1,19,97,253,455,610,630,518,338,165,55,11,1
%N A259074 Triangle T(n,k) = Sum_{j=0..(n-k)/3} C(n-3*j-1,k-1)*C(n-k-3*j,j).
%F A259074 G.f.: (x*y)/(1-x-x^4-x*y).
%e A259074 [1]
%e A259074 [1,1]
%e A259074 [1,2,1]
%e A259074 [1,3,3,1]
%e A259074 [2,4,6,4,1]
%e A259074 [3,7,10,10,5,1]
%t A259074 Table[Sum[Binomial[n - 3*j - 1, k - 1] Binomial[n - k - 3*j, j], {j, 0, (n - k)/3}], {n, 12}, {k, n}] // Flatten (* _Michael De Vlieger_, Jun 19 2015 *)
%o A259074 (Maxima) T(n,k):=sum(binomial(n-3*j-1,k-1)*binomial(n-k-3*j,j),j,0,(n-k)/3);
%Y A259074 Column k=1 gives A003269, column k=2 A292324.
%Y A259074 Row sums give A008999(n-1) for n>0.
%K A259074 nonn,tabl
%O A259074 1,5
%A A259074 _Vladimir Kruchinin_, Jun 18 2015