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A303842 Triangle read by rows: T(s,n) (s>=1 and 2<=n<=s+1) = number of trees with n nodes and positive integer edge labels with label sum s.

%I A303842 #17 May 21 2018 15:17:50
%S A303842 1,1,1,1,1,2,1,2,3,3,1,2,6,6,6,1,3,9,15,16,11,1,3,13,26,43,37,23,1,4,
%T A303842 17,46,88,116,96,47,1,4,23,68,169,273,329,239,106,1,5,28,103,287,585,
%U A303842 869,918,622,235,1,5,35,141,467,1104,2031,2695,2609,1607,551
%N A303842 Triangle read by rows: T(s,n) (s>=1 and 2<=n<=s+1) = number of trees with n nodes and positive integer edge labels with label sum s.
%H A303842 Andrew Howroyd, <a href="/A303842/b303842.txt">Table of n, a(n) for n = 1..1275</a>
%H A303842 R. J. Mathar, <a href="http://viXra.org/abs/1805.0205">Labeled Trees with fixed node label sum</a>, vixra:1805.0205 (2018).
%e A303842 The triangle starts
%e A303842 1;
%e A303842 1   1;
%e A303842 1   1   2;
%e A303842 1   2   3    3;
%e A303842 1   2   6    6    6;
%e A303842 1   3   9    15   16    11;
%e A303842 1   3   13   26   43    37     23;
%e A303842 1   4   17   46   88    116    96    47;
%e A303842 1   4   23   68   169   273    329   239  106;
%e A303842 1   5   28   103  287   585    869   918  622    235;
%e A303842 1   5   35   141  467   1104   2031  2695 2609   1607   551;
%e A303842 1   6   42   195  711   1972   4211  6882 8399   ...    4235  1301;
%e A303842 1   6   50   253  1051  3270   8108 15513 23152  ...    ... ;
%e A303842 1   7   58   330  1489  5222  14552 32191 56291  ...    ... ;
%e A303842 1   7   68   412  2063  7958  24846 62014 124958  ...    ... ;
%o A303842 (PARI)
%o A303842 EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
%o A303842 b(n)={my(v=[1]); for(i=1, n, v=concat([1], v + EulerMT(y*v))); Ser(v)*y*(1-x)}
%o A303842 seq(n)={my(g=b(n)); Vec(g + (substvec(g, [x,y], [x^2,y^2]) - g^2)*x/(2*(1-x)) - y)}
%o A303842 {my(A=seq(15)); for(n=1, #A, print(Vecrev(A[n]/y^2)))} \\ _Andrew Howroyd_, May 20 2018
%Y A303842 Cf. A303841 (labeled nodes), A000055 (diagonal), A027852 (subdiagonal), A303833 (subdiagonal), A304914 (row sums).
%K A303842 nonn,tabl
%O A303842 1,6
%A A303842 _R. J. Mathar_, May 01 2018