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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
OFFSET
1,6
LINKS
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
KEYWORD
nonn,tabl
AUTHOR
R. J. Mathar, May 01 2018
STATUS
approved