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

0,2

Links

Table of n, a(n) for n=0..44.

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

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.

Example

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],
   ...

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.

Sequence in context: A248827 A273525 A377853 * A274336 A192648 A219339

Adjacent sequences: A383450 A383451 A383452 * A383454 A383455 A383456

Keyword

nonn,tabl

Author

N. J. A. Sloane, May 02 2025

Status

approved