A395262 Triangle read by rows: Expansion of g.f. A(x,y) that satisfies the functional equation A(x,y) = Sum_{j>=0} x^(j*(j+1)/2) * y^j * A(x,y)^j / Product_{k=1..j} (1 - x^k * y * A(x,y)).

1, 0, 1, 0, 0, 2, 0, 0, 1, 5, 0, 0, 0, 4, 14, 0, 0, 0, 1, 15, 42, 0, 0, 0, 1, 7, 56, 132, 0, 0, 0, 0, 6, 36, 210, 429, 0, 0, 0, 0, 1, 32, 168, 792, 1430, 0, 0, 0, 0, 1, 12, 159, 750, 3003, 4862, 0, 0, 0, 0, 1, 7, 86, 758, 3267, 11440, 16796, 0, 0, 0, 0, 0, 7, 48, 513, 3522, 14014, 43758, 58786

Offset

0,6

Links

Table of n, a(n) for n=0..77.

Example

Triangle begins:
     k=0  1  2  3   4   5    6    7     8     9
  n=0 [1]
  n=1 [0, 1]
  n=2 [0, 0, 2]
  n=3 [0, 0, 1, 5]
  n=4 [0, 0, 0, 4, 14]
  n=5 [0, 0, 0, 1, 15, 42]
  n=6 [0, 0, 0, 1,  7, 56, 132]
  n=7 [0, 0, 0, 0,  6, 36, 210, 429]
  n=8 [0, 0, 0, 0,  1, 32, 168, 792, 1430]
  n=9 [0, 0, 0, 0,  1, 12, 159, 750, 3003, 4862]
  ...

Prog

(PARI)
tri(n) = {n*(n+1)/2}
A_xy(N) = {my(x='x+O('x^(N+1)), A=1); for(i=1, N, A = sum(j=0, N, (x^tri(j) * y^j * A^j)/prod(k=1, j, (1 - x^k*y*A)))); vector(N, n, Vecrev(polcoeff(A, n-1)))}

Crossrefs

Cf. A000108 (main diagonal), A001003 (column sums), A393488 (row sums).

Sequence in context: A361290 A285638 A325667 * A395304 A067310 A369321

Adjacent sequences: A395259 A395260 A395261 * A395263 A395264 A395265

Keyword

nonn,tabl

Author

John Tyler Rascoe, Apr 17 2026

Status

approved