A393235 - OEIS

A393235

Triangle read by rows: T(n,k) is the number of ordered rooted trees with 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 no two adjacent sibling nodes have the same weight.

1, 0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 5, 10, 1, 0, 1, 8, 20, 20, 1, 0, 1, 9, 41, 68, 35, 1, 0, 1, 12, 60, 166, 196, 56, 1, 0, 1, 13, 90, 321, 588, 490, 84, 1, 0, 1, 16, 121, 544, 1435, 1888, 1092, 120, 1, 0, 1, 17, 160, 870, 2836, 5580, 5538, 2220, 165, 1

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OFFSET

0,9

LINKS

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

FORMULA

G.f.: C(x,y) satisfies C(x,y) = 1/(1 - y*C(x,y) * Sum_{k>0} (x^k/(1 + x^k*y*C(x,y)))).

EXAMPLE

Triangle begins:

k=0 1 2 3 4 5 6 7 8

n=0 [1]

n=1 [0, 1]

n=2 [0, 1, 1]

n=3 [0, 1, 4, 1]

n=4 [0, 1, 5, 10, 1]

n=5 [0, 1, 8, 20, 20, 1]

n=6 [0, 1, 9, 41, 68, 35, 1]

n=7 [0, 1, 12, 60, 166, 196, 56, 1]

n=8 [0, 1, 13, 90, 321, 588, 490, 84, 1]

...

T(3,1) = 1: o

(3)

T(3,2) = 4: o o o o

/ \ / \ | |

(2) (1) (1) (2) (2) (1)

| |

(1) (2)

T(3,3) = 1: o

(1)

PROG

(PARI) C_xy(N) = {my(x='x+O('x^(N+1)), A=1); for(i=1, N, A= 1/(1 - A*y*sum(k=1, N, x^k/(1 + x^k*y*A)))); vector(N-1, n, Vecrev(polcoeff(A, n-1)))}

CROSSREFS

Cf. A000292 (empirical 2nd diagonal), A394411 (row sums).

Cf. A000108, A002212, A394284, A394385.

Sequence in context: A253011 A290456 A010639 * A035588 A318623 A332055

Adjacent sequences: A393232 A393233 A393234 * A393236 A393237 A393238

KEYWORD

nonn,tabl

AUTHOR

John Tyler Rascoe, Mar 19 2026

STATUS

approved