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A271362 Number T(n,k) of series-reduced free trees with n nodes of which exactly k>=3 are leaves, k+1 <= n <= 2k-2.

%I A271362 #36 Nov 14 2024 09:50:37
%S A271362 1,1,1,1,1,1,1,2,1,2,2,1,2,4,3,1,4,6,3,1,2,10,9,4,1,8,17,12,4,1,4,22,
%T A271362 30,16,5,1,15,47,44,20,5,1,6,53,91,67,25,6,1,32,127,158,91,30,6,1,11,
%U A271362 121,282,258,126
%N A271362 Number T(n,k) of series-reduced free trees with n nodes of which exactly k>=3 are leaves, k+1 <= n <= 2k-2.
%C A271362 The length of row n is floor((n-2)/2).
%H A271362 Washington Bomfim, <a href="/A271362/b271362.txt">Table of n, a(n) for n = 4..199</a>
%H A271362 B. D. McKay, <a href="http://users.cecs.anu.edu.au/~bdm/data/trees.html">Lists of Trees sorted by diameter and Homeomorphically irreducible trees, with <= 22 nodes.</a>
%F A271362 T(n,k) = A271205(k,n).
%e A271362     Irregular triangle begins
%e A271362     n \ k 3   4   5   6   7  8
%e A271362      4     1;
%e A271362      5     1;
%e A271362      6     1,  1;
%e A271362      7     1,  1;
%e A271362      8     1,  2,  1;
%e A271362      9     2,  2,  1;
%e A271362     10     2,  4,  3,  1;
%e A271362     11     4,  6,  3,  1;
%e A271362     12     2, 10,  9,  4, 1;
%e A271362     13     8, 17, 12,  4, 1;
%e A271362     14     4, 22, 30, 16, 5, 1;
%e A271362     15    15, 47, 44, 20, 5, 1;
%e A271362     ...
%o A271362 (PARI)  \\ using files hitree4.txt etc from McKay.
%o A271362 nL(n, Tr) = { my(E = strsplit(Tr, "  "), u_v, Deg = vectorsmall(n));
%o A271362 for(j = 1, n-1, u_v = strsplit(E[j], " "); u_v = eval(u_v);
%o A271362    Deg[ u_v[1]+1 ]++; Deg[ u_v[2]+1 ]++); sum(v = 1, n, Deg[v] == 1)
%o A271362 };
%o A271362 Rows(r1, r2) = {my(F, C, nF); for(n = r1, r2,
%o A271362 F = readstr(Str("hitree", n, ".txt")); C = vectorsmall(n-1);
%o A271362 for(i = 1, #F, nF = nL(n, F[i]); C[nF]++ );
%o A271362 print1(n" "); for(i=1, #C, if(C[i] > 0, print1(C[i]", "))); print() )
%o A271362 }; \\ _Washington Bomfim_, Jul 09 2021
%Y A271362 Transpose of A271205.
%Y A271362 Cf. A000014 (row sums), A345971.
%K A271362 nonn,tabf
%O A271362 4,8
%A A271362 _Stephan Beyer_, Apr 05 2016