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A343648 Triangle read by rows, 1 <= k <= n: T(n,k) is the number of (unlabeled) connected graphs with n nodes and zero forcing number k.

%I A343648 #9 Apr 26 2021 21:32:17
%S A343648 1,1,0,1,1,0,1,4,1,0,1,10,9,1,0,1,33,58,19,1,0,1,94,457,266,34,1,0,1,
%T A343648 319,3977,5574,1184,61,1,0,1,1053,39547,142039,72944,5393,102,1,0,1,
%U A343648 3683,414891,4170606,5919941,1180610,26668,170,1,0
%N A343648 Triangle read by rows, 1 <= k <= n: T(n,k) is the number of (unlabeled) connected graphs with n nodes and zero forcing number k.
%C A343648 The zero forcing number of a graph can be defined as follows. Start with a blue/white coloring of the nodes. At each step, all white nodes, which are currently the unique white neighbor of a blue node, are colored blue. The zero forcing number is the minimum number of blue nodes in an initial coloring that leads to all nodes being blue after a finite number of steps.
%H A343648 Shaun M. Fallat, Leslie Hogben, Jephian C.-H. Lin, and Bryan L. Shader, <a href="https://www.ams.org/journals/notices/202002/rnoti-p257.pdf">The inverse eigenvalue problem of a graph, zero forcing, and related parameters</a>, Notices of the American Mathematical Society 67 (2020), 257-261.
%F A343648 T(n,1) = 1. (The path graph is the only n-node graph with zero forcing number 1.)
%F A343648 T(n,n-1) = 1 for n >= 2. (The complete graph is the only connected n-node graph with zero forcing number n-1.)
%F A343648 T(n,n) = 0 for n >= 2.
%e A343648 Triangle begins:
%e A343648    n\k 1     2       3        4        5        6      7    8  9  10
%e A343648   ------------------------------------------------------------------
%e A343648    1:  1
%e A343648    2:  1     0
%e A343648    3:  1     1       0
%e A343648    4:  1     4       1        0
%e A343648    5:  1    10       9        1        0
%e A343648    6:  1    33      58       19        1        0
%e A343648    7:  1    94     457      266       34        1      0
%e A343648    8:  1   319    3977     5574     1184       61      1    0
%e A343648    9:  1  1053   39547   142039    72944     5393    102    1  0
%e A343648   10:  1  3683  414891  4170606  5919941  1180610  26668  170  1   0
%Y A343648 Row sums: A001349.
%Y A343648 Cf. A343649.
%K A343648 nonn,tabl
%O A343648 1,8
%A A343648 _Pontus von Brömssen_, Apr 24 2021