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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. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified June 17 18:04 EDT 2019. Contains 324198 sequences. (Running on oeis4.)