A383453 The Geode Bi-Tri infinite rectangular array, read by upward antidiagonals.

1, 2, 3, 5, 16, 12, 14, 70, 110, 55, 42, 288, 702, 728, 273, 132, 1155, 3850, 6160, 4760, 1428, 429, 4576, 19448, 42432, 50388, 31008, 7752, 1430, 18018, 93366, 259350, 418950, 395010, 201894, 43263, 4862, 70720, 433160, 1466080, 3010700, 3853696, 3010700, 1315600, 246675

Offset: 0

Author

N. J. A. Sloane, May 02 2025

Keywords

• nonn
• tabl

Examples

			The array begins:
      1,     3,      12,       55,       273,        1428, ...
      2,    16,     110,      728,      4760,       31008, ...
      5,    70,     702,     6160,     50388,      395010, ...
     14,   288,    3850,    42432,    418950,     3853696, ...
     42,  1155,   19448,   259350,   3010700,    31870410, ...
    132,  4576,   93366,  1466080,  19612560,   235282320, ...
    429, 18018,  433160,  7845024, 119041650,  1598394798, ...
   1430, 70720, 1961256, 40310400, 685026342, 10189625600, ...
   ...
The first few antidiagonals are:
   [  1],
   [  2,    3],
   [  5,   16,    12],
   [ 14,   70,   110,    55],
   [ 42,  288,   702,   728,   273],
   [132, 1155,  3850,  6160,  4760,  1428],
   [429, 4576, 19448, 42432, 50388, 31008, 7752],
   ...
		

Links

• N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025).

Crossrefs

Same table as A243660, transposed. - Dan Eilers, May 06 2025

The leading row and column of the rectangular array are A001764 and A000108.

See A383454-A383457 for other rows and columns.

Formula

The entry (m2, m3) of the rectangular array is equal to (2*m2 + 3*m3 + 3)!/((2*m2 + 2*m3 + 3)*(m2 + m3 + 1)*(m2 + 2*m3 + 2)! * m2! * m3!), with m2, m3 >= 0.