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Triangle T(n,k) read by rows: the number of independent sets of size k in the 132 core of size n.
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%I #17 Sep 28 2019 12:30:28

%S 1,1,1,1,3,3,1,6,14,16,1,10,40,85,105,1,15,90,295,594,771,1,21,175,

%T 805,2331,4529,6083,1,28,308,1876,7280,19348,36644,50464,1,36,504,

%U 3906,19404,66780,166608,309537,434493,1,45,780,7470,45990,197484,621180,1476135,2701610,3849715

%N Triangle T(n,k) read by rows: the number of independent sets of size k in the 132 core of size n.

%H C. Bean, M. Tannock, H. Ulfarsson, <a href="http://arxiv.org/abs/1512.08155">Pattern avoiding permutations and independent sets in graphs</a>, arXiv:1512.08155 [math.CO], 2015, Theorem 3.5.

%F The bivariate g.f. G(x,y) satisfies G = 1+x*G+x*y*G^2/(1-y*(G-1)).

%F n*T(n,k) = Sum_{j=0..n-1} binomial(n,k-j)*binomial(n,j+1)*binomial(n-1+j,n-1).

%e 1;

%e 1, 1;

%e 1, 3, 3;

%e 1, 6, 14, 16;

%e 1, 10, 40, 85, 105;

%e 1, 15, 90, 295, 594, 771;

%e 1, 21, 175, 805, 2331, 4529, 6083;

%e 1, 28, 308, 1876, 7280, 19348, 36644, 50464;

%e 1, 36, 504, 3906, 19404, 66780, 166608, 309537, 434493;

%t T[n_, k_] := Binomial[n-1, k] HypergeometricPFQ[{-k, 2-n, n-1}, {2, n-k}, 1];

%t Table[T[n, k], {n, 1, 10}, {k, 0, n-1}] (* _Jean-François Alcover_, Sep 28 2019 *)

%K easy,tabl,nonn

%O 1,5

%A _R. J. Mathar_, Nov 20 2016