A394386 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 non-root nodes of odd weight can have branches.

1, 0, 1, 0, 1, 2, 0, 1, 3, 5, 0, 1, 5, 10, 14, 0, 1, 6, 21, 35, 42, 0, 1, 8, 31, 86, 126, 132, 0, 1, 9, 48, 150, 350, 462, 429, 0, 1, 11, 63, 261, 700, 1422, 1716, 1430, 0, 1, 12, 86, 390, 1345, 3192, 5775, 6435, 4862, 0, 1, 14, 106, 584, 2241, 6692, 14322, 23452, 24310, 16796

Offset

0,6

Links

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

Formula

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

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,  2]
 n=3 [0, 1,  3,  5]
 n=4 [0, 1,  5, 10,  14]
 n=5 [0, 1,  6, 21,  35,   42]
 n=6 [0, 1,  8, 31,  86,  126,  132]
 n=7 [0, 1,  9, 48, 150,  350,  462,  429]
 n=8 [0, 1, 11, 63, 261,  700, 1422, 1716, 1430]
 n=9 [0, 1, 12, 86, 390, 1345, 3192, 5775, 6435, 4862]
 ...
T(2,1) = 1:  o
             |
            (2)
T(2,2) = 2:  o      o
             |     / \
            (1)  (1) (1)
             |
            (1)

Prog

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

Crossrefs

Cf. A000108 (main diagonal), A001700 (empirical 2nd diagonal).

Cf. A002212, A394284.

Sequence in context: A367562 A213861 A355173 * A108458 A254281 A295682

Adjacent sequences: A394383 A394384 A394385 * A394387 A394388 A394389

Keyword

nonn,tabl,easy

Author

John Tyler Rascoe, Mar 19 2026

Status

approved