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.

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, 17, 46, 88, 116, 96, 47, 1, 4, 23, 68, 169, 273, 329, 239, 106, 1, 5, 28, 103, 287, 585, 869, 918, 622, 235, 1, 5, 35, 141, 467, 1104, 2031, 2695, 2609, 1607, 551

Offset: 1

Author

R. J. Mathar, May 01 2018

Keywords

• nonn
• tabl

Examples

			The triangle starts
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   17   46   88    116    96    47;
1   4   23   68   169   273    329   239  106;
1   5   28   103  287   585    869   918  622    235;
1   5   35   141  467   1104   2031  2695 2609   1607   551;
1   6   42   195  711   1972   4211  6882 8399   ...    4235  1301;
1   6   50   253  1051  3270   8108 15513 23152  ...    ... ;
1   7   58   330  1489  5222  14552 32191 56291  ...    ... ;
1   7   68   412  2063  7958  24846 62014 124958  ...    ... ;
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275
• R. J. Mathar, Labeled Trees with fixed node label sum, vixra:1805.0205 (2018).

Crossrefs

Cf. A303841 (labeled nodes), A000055 (diagonal), A027852 (subdiagonal), A303833 (subdiagonal), A304914 (row sums).

Programs

PARI

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)}
b(n)={my(v=[1]); for(i=1, n, v=concat([1], v + EulerMT(y*v))); Ser(v)*y*(1-x)}
seq(n)={my(g=b(n)); Vec(g + (substvec(g, [x,y], [x^2,y^2]) - g^2)*x/(2*(1-x)) - y)}
{my(A=seq(15)); for(n=1, #A, print(Vecrev(A[n]/y^2)))} \\ Andrew Howroyd, May 20 2018