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 A289483 Number of gcds-sortable two-rooted graphs on n vertices such that all vertices have even degree. 0
 0, 1, 1, 5, 29, 365, 7565, 259533, 16766541, 1695913805, 319025518925, 99428910374221, 53629954918196557, 51436455420773021005, 81633965668282476025165, 234346782219278654389392717, 1131832076434284133556933170509 (list; graph; refs; listen; history; text; internal format)
 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 Table of n, a(n) for n=1..17. 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 Cf. A289472. Sequence in context: A226666 A366411 A356486 * A216027 A087899 A202759 Adjacent sequences: A289480 A289481 A289482 * A289484 A289485 A289486 KEYWORD nonn AUTHOR Sam Heil, Jul 06 2017 STATUS approved

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Last modified June 20 03:21 EDT 2024. Contains 373512 sequences. (Running on oeis4.)