

A289472


Number of gcdssortable tworooted graphs on n vertices.


1



0, 1, 1, 17, 113, 7729, 224689, 61562033, 7309130417, 8013328398001, 3825133597372081, 16776170217003753137, 32072986971771549318833, 562672074981014060438175409, 4304275145962667488546071527089, 302049699050029408242290021253725873
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OFFSET

1,4


COMMENTS

This formula comes from the fact that for each possible value of the (n2)vertex subgraph G containing all of the nonroot vertices, if G has adjacency matrix A over F_2 then there are 4^rank(A) tworooted gcdssortable graphs containing the nonroot subgraph G. We can apply the formula from MacWilliams for the number of symmetric binary matrices with zero diagonal of each rank to get the total number of gcdssortable graphs.


LINKS

Table of n, a(n) for n=1..16.
C. A. Brown, C. S. Carrillo Vazquez, R. Goswami, S. Heil, and M. Scheepers, The Sortability of Graphs and Matrices Under Context Directed Swaps
F. J. MacWilliams, Orthogonal matrices over finite fields, Amer. Math. Monthly, 76 (1969), 152164.


FORMULA

a(n) = Sum_{s=0..floor(n/2)1} 2^(s^2+3s) * (Product_{i=0..2s1} (2^(n2i)1) / Product_{i=1..s} (2^(2i)1)).


MATHEMATICA

Table[Sum[2^(s^2 + 3 s) (Product[(2^(n  2  i)  1), {i, 0, 2 s  1}]/Product[(2^(2 i)  1), {i, s}]), {s, 0, Floor[n/2]  1}], {n, 16}] (* Michael De Vlieger, Jul 30 2017 *)


PROG

(PARI) a(n) = sum(s=0, n\21, 2^(s^2+3*s)*prod(i=0, 2*s1, (2^(n2i)1))/prod(i=1, s, 2^(2*i)1)); \\ Michel Marcus, Jul 07 2017


CROSSREFS

Sequence in context: A105127 A142403 A197360 * A172521 A226688 A070144
Adjacent sequences: A289469 A289470 A289471 * A289473 A289474 A289475


KEYWORD

nonn


AUTHOR

Sam Heil, Jul 06 2017


STATUS

approved



