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A303841 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.

%I A303841 #28 Nov 21 2018 02:33:12
%S A303841 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 A303841 30,50,63,48,23,1,4,16,44,96,146,164,115,47,1,5,20,67,164,315,437,444,
%U A303841 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 A303841 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 A303841 Andrew Howroyd, <a href="/A303841/b303841.txt">Table of n, a(n) for n = 1..1275</a>
%H A303841 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 A303841 R. J. Mathar, <a href="http://vixra.org/abs/1805.0205">Labeled trees with fixed node label sum</a> vixra:1805.0205 (2018).
%H A303841 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 A303841 The triangle starts
%e A303841 1;
%e A303841 1 1;
%e A303841 1 1  1;
%e A303841 1 2  2   2;
%e A303841 1 2  4   4    3;
%e A303841 1 3  6  10    9   6;
%e A303841 1 3  9  17   24   20    11;
%e A303841 1 4 12  30   50   63    48    23;
%e A303841 1 4 16  44   96  146   164   115    47;
%e A303841 1 5 20  67  164  315   437   444   286  106;
%e A303841 1 5 25  91  267  592  1022  1300  1204  719     235;
%e A303841 1 6 30 126  408 1059  2126  3331  3899 3328    1842    551;
%e A303841 1 6 36 163  603 1754  4098  7511 10781 11692   9233   4766  1301;
%e A303841 1 7 42 213  856 2805  7368 15619 26294 34844  35136  25865 12486  3159;
%e A303841 1 7 49 265 1186 4270 12590 30111 58485 91037 112036 105592 72734 32973 7741;
%o A303841 (PARI) \\ here b is A303911
%o A303841 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 A303841 b(n)={my(v=[1]); for(i=2, n, v=concat([1], v + EulerMT(y*v))); v}
%o A303841 seq(n)={my(g=x*Ser(y*b(n))); Vec(g - g^2/2 + substvec(g,[x,y],[x^2,y^2])/2)}
%o A303841 {my(A=seq(15)); for(n=1, #A, print(Vecrev(A[n]/y)))} \\ _Andrew Howroyd_, May 19 2018
%Y A303841 Cf. A036250 (row sums), A002620 (column 3), A301739 (column 4), A301740 (column 5), A000055 (diagonal), A000081 (subdiagonal), A303911 (rooted).
%K A303841 nonn,tabl
%O A303841 1,8
%A A303841 _R. J. Mathar_, May 01 2018