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

1,6

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).

Example

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  ...    ... ;

Prog

(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

Crossrefs

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

Sequence in context: A227909 A301984 A210805 * A057041 A267177 A099567

Adjacent sequences: A303839 A303840 A303841 * A303843 A303844 A303845

Keyword

nonn,tabl

Author

R. J. Mathar, May 01 2018

Status

approved