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.

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, 30, 50, 63, 48, 23, 1, 4, 16, 44, 96, 146, 164, 115, 47, 1, 5, 20, 67, 164, 315, 437, 444, 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

Offset

1,8

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275

F. Harary, G. Prins, The number of homeomorphically irreducible trees and other species, Acta. Math. 101 (1959) 141, equation (9b).

R. J. Mathar, Labeled trees with fixed node label sum vixra:1805.0205 (2018).

Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.

Example

The triangle starts
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  30   50   63    48    23;
1 4 16  44   96  146   164   115    47;
1 5 20  67  164  315   437   444   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;
1 6 36 163  603 1754  4098  7511 10781 11692   9233   4766  1301;
1 7 42 213  856 2805  7368 15619 26294 34844  35136  25865 12486  3159;
1 7 49 265 1186 4270 12590 30111 58485 91037 112036 105592 72734 32973 7741;

Prog

(PARI) \\ here b is A303911
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=2, n, v=concat([1], v + EulerMT(y*v))); v}
seq(n)={my(g=x*Ser(y*b(n))); Vec(g - g^2/2 + substvec(g, [x, y], [x^2, y^2])/2)}
{my(A=seq(15)); for(n=1, #A, print(Vecrev(A[n]/y)))} \\ Andrew Howroyd, May 19 2018

Crossrefs

Cf. A036250 (row sums), A002620 (column 3), A301739 (column 4), A301740 (column 5), A000055 (diagonal), A000081 (subdiagonal), A303911 (rooted).

Sequence in context: A253196 A353981 A271205 * A093116 A327799 A198199

Adjacent sequences: A303838 A303839 A303840 * A303842 A303843 A303844

Keyword

nonn,tabl

Author

R. J. Mathar, May 01 2018

Status

approved