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A380363 Triangle read by rows: T(n,k) is the number of linear trees with n vertices and k vertices of degree >= 3, 0 <= k <= max(0, floor(n/2)-1).

%I A380363 #16 Sep 05 2025 09:27:55
%S A380363 1,1,1,1,1,1,1,2,1,4,1,1,7,3,1,11,10,1,1,17,24,5,1,25,56,22,1,1,36,
%T A380363 114,74,6,1,50,224,219,37,1,1,70,411,576,158,8,1,94,733,1394,591,58,1,
%U A380363 1,127,1252,3150,1896,304,9,1,168,2091,6733,5537,1342,82,1
%N A380363 Triangle read by rows: T(n,k) is the number of linear trees with n vertices and k vertices of degree >= 3, 0 <= k <= max(0, floor(n/2)-1).
%C A380363 A linear tree is a tree with all vertices of degree > 2 belonging to a single path. These are equinumerous with lobster graphs. All trees having at most 3 vertices of degree > 2 are linear trees.
%H A380363 Andrew Howroyd, <a href="/A380363/b380363.txt">Table of n, a(n) for n = 0..2501</a> (rows 0..100)
%H A380363 Tanay Wakhare, Eric Wityk, and Charles R. Johnson, <a href="https://doi.org/10.1016/j.disc.2020.112008">The proportion of trees that are linear</a>, Discrete Mathematics 343.10 (2020): 112008. Also <a href="https://doi.org/10.1016/j.disc.2020.112284">Corrigendum</a> and preprint <a href="https://arxiv.org/abs/1901.08502">arXiv:1901.08502</a>. See Table 1.
%H A380363 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LobsterGraph.html">Lobster Graph</a>.
%e A380363 Triangle begins:
%e A380363   1;
%e A380363   1;
%e A380363   1;
%e A380363   1;
%e A380363   1,   1;
%e A380363   1,   2;
%e A380363   1,   4,    1;
%e A380363   1,   7,    3;
%e A380363   1,  11,   10,    1;
%e A380363   1,  17,   24,    5;
%e A380363   1,  25,   56,   22,    1;
%e A380363   1,  36,  114,   74,    6;
%e A380363   1,  50,  224,  219,   37,   1;
%e A380363   1,  70,  411,  576,  158,   8;
%e A380363   1,  94,  733, 1394,  591,  58, 1;
%e A380363   1, 127, 1252, 3150, 1896, 304, 9;
%e A380363   ...
%o A380363 (PARI)
%o A380363 G(n,y)={my(p=1/eta(x + O(x^n)), p2=1/eta(x^2 + O(x^n)),
%o A380363   g1=(p - 1/(1-x))^2/((1 - x)*(1 - x*y*(p-1)/(1-x))),
%o A380363   g2=(p2 - 1/(1-x^2))*(1 + x + x*y*(p-1))/((1 - x^2)*(1 - x^2*y^2*(p2-1)/(1-x^2))) );
%o A380363   x^2*y^2*(g1 + g2)/2 + x*y*(p - 1/((1 + x)*(1 - x)^2)) + 1/(1-x)
%o A380363 }
%o A380363 T(n)=[Vecrev(p) | p<-Vec(G(n,y))]
%o A380363 {my(A=T(15)); for(i=1, #A, print(A[i]))}
%Y A380363 Columns 0..4 are A000012, A004250(n-1), A338706, A338707, A338708.
%Y A380363 Row sums are A130131.
%Y A380363 Cf. A238415 (initial columns same up to k=3).
%K A380363 nonn,tabf,changed
%O A380363 0,8
%A A380363 _Andrew Howroyd_, Jan 26 2025