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

1,5

Links

Table of n, a(n) for n=1..55.

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

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).

Example

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;

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 *)

Crossrefs

Sequence in context: A094040 A039798 A193560 * A356916 A001498 A240439

Adjacent sequences: A278387 A278388 A278389 * A278391 A278392 A278393

Keyword

easy,tabl,nonn

Author

R. J. Mathar, Nov 20 2016

Status

approved