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 A303911 Triangle T(w>=1,1<=n<=w) read by rows: the number of rooted weighted trees with n nodes and weight w. 3
 1, 1, 1, 1, 2, 2, 1, 3, 5, 4, 1, 4, 10, 13, 9, 1, 5, 16, 31, 35, 20, 1, 6, 24, 60, 98, 95, 48, 1, 7, 33, 103, 217, 304, 262, 115, 1, 8, 44, 162, 423, 764, 945, 727, 286, 1, 9, 56, 241, 743, 1658, 2643, 2916, 2033, 719, 1, 10, 70, 341, 1221, 3224, 6319, 8996, 8984, 5714, 1842, 1, 11, 85, 466, 1893 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Weights are positive integer labels on the nodes. The weight of the tree is the sum of the weights of its nodes. 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) 141-162, W(x,y) equation (9a) EXAMPLE The triangle starts 1 ; 1  1 ; 1  2  2 ; 1  3  5   4 ; 1  4 10  13    9 ; 1  5 16  31   35    20 ; 1  6 24  60   98    95    48 ; 1  7 33 103  217   304   262   115 ; The first column (for a single node n=1) is 1, because all the weight is on that node. 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)} seq(n)={my(v=); for(i=2, n, v=concat(, v + EulerMT(y*v))); v} {my(A=seq(10)); for(n=1, #A, print(Vecrev(A[n])))} \\ Andrew Howroyd, May 19 2018 CROSSREFS Cf. A000081 (diagonal), A000107 (subdiagonal), A036249 (row sums), A303841 (not rooted). Sequence in context: A026300 A099514 A228352 * A205575 A257006 A139687 Adjacent sequences:  A303908 A303909 A303910 * A303912 A303913 A303914 KEYWORD nonn,tabl AUTHOR R. J. Mathar, May 02 2018 STATUS approved

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Last modified May 28 17:37 EDT 2020. Contains 334684 sequences. (Running on oeis4.)