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