A382439 Triangle read by rows: defined by the two-variable g.f. (x^3*y^2 + x^3*y - x^2*y + 1) / (1 - x^2*y - x*y - x).

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 7, 12, 7, 1, 1, 9, 24, 24, 9, 1, 1, 11, 40, 60, 40, 11, 1, 1, 13, 60, 124, 124, 60, 13, 1, 1, 15, 84, 224, 308, 224, 84, 15, 1, 1, 17, 112, 368, 656, 656, 368, 112, 17, 1, 1, 19, 144, 564, 1248, 1620, 1248, 564, 144, 19, 1

Offset: 0

Author

F. Chapoton, Mar 25 2025

Keywords

• nonn
• tabl

Comments

The alternating sum of every row n > 0 vanishes. Every row is symmetric.

Examples

			  [0] [1]
  [1] [1,  1]
  [2] [1,  2,   1]
  [3] [1,  5,   5,   1]
  [4] [1,  7,  12,   7,   1]
  [5] [1,  9,  24,  24,   9,   1]
  [6] [1, 11,  40,  60,  40,  11,   1]
  [7] [1, 13,  60, 124, 124,  60,  13,   1]
  [8] [1, 15,  84, 224, 308, 224,  84,  15,  1]
  [9] [1, 17, 112, 368, 656, 656, 368, 112, 17, 1]
		

Crossrefs

Similar to A008288 and A382436. Row sums are A245990.

Programs

Sage

y = polygen(QQ, 'y')
x = y.parent()[['x']].gen()
gf = (x^3*y^2 + x^3*y - x^2*y + 1)/(1 - x^2*y - x*y - x)
[list(u) for u in list(gf.O(10))]