OFFSET
1,4
COMMENTS
This formula comes from the fact that for each possible value of the (n-2)-vertex subgraph G containing all of the non-root vertices, if G has adjacency matrix A over F_2 then there are 2^rank(A) two-rooted gcds-sortable graphs with all vertices of even degree containing the non-root subgraph G. Then, 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 gcds-sortable graphs with all vertices of even degree.
LINKS
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), 152-164.
FORMULA
a(n) = Sum_{s=0..floor(n/2)-1} 2^((s^2+3s)/2) * (Product_{i=0..2s-1} (2^(n-2-i)-1) / Product_{i=1..s} (2^(2i)-1))
MATHEMATICA
Table[Sum[2^((s^2 + 3 s)/2) * Product[(2^(n - 2 - i) - 1), {i, 0, 2 s - 1}]/Product[(2^(2 j) - 1), {j, s}], {s, 0, Floor[n/2] - 1}], {n, 2, 17}] (* Michael De Vlieger, Jul 12 2017 *)
PROG
(PARI) a(n) = sum(s=0, n\2-1, 2^((s^2+3*s)/2)*prod(i=0, 2*s-1, (2^(n-2-i)-1))/prod(i=1, s, 2^(2*i)-1)); \\ Michel Marcus, Jul 07 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Sam Heil, Jul 06 2017
STATUS
approved