A278390 Triangle T(n,k) read by rows: the number of independent sets of size k in the 132 core of size n.

1, 1, 1, 1, 3, 3, 1, 6, 14, 16, 1, 10, 40, 85, 105, 1, 15, 90, 295, 594, 771, 1, 21, 175, 805, 2331, 4529, 6083, 1, 28, 308, 1876, 7280, 19348, 36644, 50464, 1, 36, 504, 3906, 19404, 66780, 166608, 309537, 434493, 1, 45, 780, 7470, 45990, 197484, 621180, 1476135, 2701610, 3849715

Offset: 1

Author

R. J. Mathar, Nov 20 2016

Keywords

• easy
• tabl
• nonn

Examples

			1;
1,  1;
1,  3,   3;
1,  6,  14,   16;
1, 10,  40,   85,   105;
1, 15,  90,  295,   594,   771;
1, 21, 175,  805,  2331,  4529,   6083;
1, 28, 308, 1876,  7280, 19348,  36644,  50464;
1, 36, 504, 3906, 19404, 66780, 166608, 309537, 434493;
		

Links

• C. Bean, M. Tannock, H. Ulfarsson, Pattern avoiding permutations and independent sets in graphs, arXiv:1512.08155 [math.CO], 2015, Theorem 3.5.

Programs

Mathematica

T[n_, k_] := Binomial[n-1, k] HypergeometricPFQ[{-k, 2-n, n-1}, {2, n-k}, 1];
Table[T[n, k], {n, 1, 10}, {k, 0, n-1}] (* Jean-François Alcover, Sep 28 2019 *)

Formula

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

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