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Triangle read by rows: T(s,n) (s>=1 and 1<=n<=s) = number of weighted trees with n nodes and positive integer node labels with label sum s.
5

%I #28 Nov 21 2018 02:33:12

%S 1,1,1,1,1,1,1,2,2,2,1,2,4,4,3,1,3,6,10,9,6,1,3,9,17,24,20,11,1,4,12,

%T 30,50,63,48,23,1,4,16,44,96,146,164,115,47,1,5,20,67,164,315,437,444,

%U 286,106,1,5,25,91,267,592,1022,1300,1204,719,235,1,6,30,126,408,1059,2126,3331,3899,3328,1842,551

%N Triangle read by rows: T(s,n) (s>=1 and 1<=n<=s) = number of weighted trees with n nodes and positive integer node labels with label sum s.

%H Andrew Howroyd, <a href="/A303841/b303841.txt">Table of n, a(n) for n = 1..1275</a>

%H F. Harary, G. Prins, <a href="http://dx.doi.org/10.1007/BF02559543">The number of homeomorphically irreducible trees and other species</a>, Acta. Math. 101 (1959) 141, equation (9b).

%H R. J. Mathar, <a href="http://vixra.org/abs/1805.0205">Labeled trees with fixed node label sum</a> vixra:1805.0205 (2018).

%H Richard J. Mathar, <a href="https://arxiv.org/abs/1808.06264">Counting Connected Graphs without Overlapping Cycles</a>, arXiv:1808.06264 [math.CO], 2018.

%e The triangle starts

%e 1;

%e 1 1;

%e 1 1 1;

%e 1 2 2 2;

%e 1 2 4 4 3;

%e 1 3 6 10 9 6;

%e 1 3 9 17 24 20 11;

%e 1 4 12 30 50 63 48 23;

%e 1 4 16 44 96 146 164 115 47;

%e 1 5 20 67 164 315 437 444 286 106;

%e 1 5 25 91 267 592 1022 1300 1204 719 235;

%e 1 6 30 126 408 1059 2126 3331 3899 3328 1842 551;

%e 1 6 36 163 603 1754 4098 7511 10781 11692 9233 4766 1301;

%e 1 7 42 213 856 2805 7368 15619 26294 34844 35136 25865 12486 3159;

%e 1 7 49 265 1186 4270 12590 30111 58485 91037 112036 105592 72734 32973 7741;

%o (PARI) \\ here b is A303911

%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=2, n, v=concat([1], v + EulerMT(y*v))); v}

%o seq(n)={my(g=x*Ser(y*b(n))); Vec(g - g^2/2 + substvec(g,[x,y],[x^2,y^2])/2)}

%o {my(A=seq(15)); for(n=1, #A, print(Vecrev(A[n]/y)))} \\ _Andrew Howroyd_, May 19 2018

%Y Cf. A036250 (row sums), A002620 (column 3), A301739 (column 4), A301740 (column 5), A000055 (diagonal), A000081 (subdiagonal), A303911 (rooted).

%K nonn,tabl

%O 1,8

%A _R. J. Mathar_, May 01 2018