A394284 Triangle read by rows: T(n,k) is the number of ordered rooted trees with non-root node weights summing to n that have k non-root nodes; such that the root has weight 0, non-root nodes have positive integer weights, and only nodes of even weight can have branches.

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 4, 6, 1, 0, 1, 6, 13, 10, 1, 0, 1, 7, 23, 34, 15, 1, 0, 1, 9, 36, 75, 75, 21, 1, 0, 1, 10, 51, 144, 215, 146, 28, 1, 0, 1, 12, 70, 240, 491, 546, 259, 36, 1, 0, 1, 13, 90, 376, 967, 1491, 1246, 428, 45, 1

Offset

0,9

Links

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

Formula

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

Example

Triangle begins:
     k=0  1   2   3    4    5    6    7   8  9
 n=0  [1]
 n=1  [0, 1]
 n=2  [0, 1,  1]
 n=3  [0, 1,  3,  1]
 n=4  [0, 1,  4,  6,   1]
 n=5  [0, 1,  6, 13,  10,   1]
 n=6  [0, 1,  7, 23,  34,  15,   1]
 n=7  [0, 1,  9, 36,  75,  75,  21,   1]
 n=8  [0, 1, 10, 51, 144, 215, 146,  28,  1]
 n=9  [0, 1, 12, 70, 240, 491, 546, 259, 36, 1]
 ...
T(3,1) = 1:   o
              |
             (3)
T(3,2) = 3:     o        o      o
               / \      / \     |
             (1) (2)  (2) (1)  (2)
                                |
                               (1)
T(3,3) = 1:     _o_
               / | \
             (1)(1)(1)

Prog

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

Crossrefs

Cf. A000217 (2nd diagonal), A032766 (column k=2), A394285 (row sums).

Cf. A000108, A107131, A384747.

Sequence in context: A380763 A356656 A085391 * A280880 A050143 A103495

Adjacent sequences: A394281 A394282 A394283 * A394285 A394286 A394287

Keyword

nonn,tabl,easy

Author

John Tyler Rascoe, Mar 14 2026

Status

approved