A336194 Table read by antidiagonals upwards: T(n,k) = (n - 1)*k^3 - 1, with n > 1 and k > 0.

0, 1, 7, 2, 15, 26, 3, 23, 53, 63, 4, 31, 80, 127, 124, 5, 39, 107, 191, 249, 215, 6, 47, 134, 255, 374, 431, 342, 7, 55, 161, 319, 499, 647, 685, 511, 8, 63, 188, 383, 624, 863, 1028, 1023, 728, 9, 71, 215, 447, 749, 1079, 1371, 1535, 1457, 999, 10, 79, 242, 511, 874, 1295, 1714, 2047, 2186, 1999, 1330

Offset: 2

Author

Stefano Spezia, Jul 11 2020

Keywords

• nonn
• easy
• tabl

Comments

T(n, k) is a sharp upper bound of the tree width of a graph G that does not contain a clique on n vertices nor a minimal separator of size larger than k (see Theorem 2.1 in Pilipczuk et al.).

All the square matrices starting at top left of the table T are singular except for the 2 X 2 submatrix: det([0, 7; 1, 15]) = -7.

Examples

			The table starts at row n = 2 and column k = 1 as:
0   7   26   63  124   215 ...
1  15   53  127  249   431 ...
2  23   80  191  374   647 ...
3  31  107  255  499   863 ...
4  39  134  319  624  1079 ...
5  47  161  383  749  1295 ...
...
		

Links

• Marcin Pilipczuk, Ni Luh Dewi Sintiari, Stéphan Thomassé and Nicolas Trotignon, (Theta, triangle)-free and (even hole, K4)-free graphs. Part 2 : bounds on treewidth, arXiv:2001.01607 [cs.DM], 2020. See p. 7.
• Index entries for sequences related to trees.

Crossrefs

Cf. A000578, A001093, A001477 (k = 1), A004771 (k = 2), A068601 (n = 2), A085537, A109129, A123865 (main diagonal), A325543, A325612.

Programs

Mathematica

T[n_,k_]:=(n-1)*k^3-1; Flatten[Table[T[n+1-k,k],{n,2,12},{k,1,n-1}]]

PARI

T(n, k) = (n - 1)*k^3 - 1

Formula

O.g.f.: x^2*y*(y*(7 - 2*y + y^2) + x*(1 - y)^3)/((1 - x)^2*(1 - y)^4).

E.g.f.: -1 + exp(x) - x + exp(y)*x + exp(y)*(1 + y + 3*y^2 + y^3) + exp(x + y)*(-1 +(-1 + x)*y*(1 + 3*y + y^2)).

T(n, k) = n*A000578(k) - A001093(k).

T(n, n) = A085537(n) - 1 for n > 1.

T(n, k) = T(n+1, 1)*T(2, k) + T(n, 1).