A394385 Triangle read by rows: T(n,k) is the number of ordered rooted trees with node weights summing to n that have k leaf nodes; where the root has weight 0 and non-root nodes have positive integer weights.

1, 0, 1, 0, 2, 1, 0, 4, 5, 1, 0, 8, 18, 9, 1, 0, 16, 56, 50, 14, 1, 0, 32, 160, 220, 110, 20, 1, 0, 64, 432, 840, 645, 210, 27, 1, 0, 128, 1120, 2912, 3150, 1575, 364, 35, 1, 0, 256, 2816, 9408, 13552, 9534, 3388, 588, 44, 1, 0, 512, 6912, 28800, 53088, 49644, 24822, 6636, 900, 54, 1

Offset

0,5

Links

Table of n, a(n) for n=0..65.

Formula

G.f.: (1 - x*y + (x - 1)*sqrt((x^2*(y^2 + 4) - 2*x*(y + 2) + 1)/(x - 1)^2))/(2*x).

Example

Triangle begins:
    k=0    1     2     3     4     5    6   7  8
 n=0 [1]
 n=1 [0,   1]
 n=2 [0,   2,    1]
 n=3 [0,   4,    5,    1]
 n=4 [0,   8,   18,    9,    1]
 n=5 [0,  16,   56,   50,   14,    1]
 n=6 [0,  32,  160,  220,  110,   20,   1]
 n=7 [0,  64,  432,  840,  645,  210,  27,  1]
 n=8 [0, 128, 1120, 2912, 3150, 1575, 364, 35, 1]
 ...
T(2,1) = 2:  o   o
             |   |
            (2) (1)
                 |
                (1)
T(2,2) = 1:    o
              / \
            (1) (1)

Prog

(PARI) C_xy(N) = { my(x='x+O('x^(N+2)), h = (1 - x*y + (x - 1)*sqrt((x^2*(y^2 + 4) - 2*x*(y + 2) + 1)/(x - 1)^2))/(2*x)); vector(N-1, n, Vecrev(polcoeff(h, n-1)))}

Crossrefs

Cf. A002212 (row sums).

Cf. A000108, A394284.

Sequence in context: A100887 A073592 A164994 * A269952 A361954 A342500

Adjacent sequences: A394382 A394383 A394384 * A394386 A394387 A394388

Keyword

nonn,tabl,easy

Author

John Tyler Rascoe, Mar 18 2026

Status

approved