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
Examples
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;
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.
Crossrefs
Programs
-
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