A253217 Number of n X n nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 2 and every value within 2 of its king move distance from the upper left and every value increasing by 0 or 1 with every step right, diagonally se or down.

0, 0, 1, 19, 268, 3568, 47698, 649712, 9023385, 127419681, 1823918697, 26398702645, 385582981615, 5674890516295, 84060883775765, 1252066289632643, 18738613233957420, 281620474177057788, 4248088188086420832

Offset

1,4

Comments

Diagonal of A253223.

Links

R. H. Hardin, Table of n, a(n) for n = 1..37

Robert Dougherty-Bliss, Experimental Methods in Number Theory and Combinatorics, Ph. D. Dissertation, Rutgers Univ. (2024). See p. 29.

Robert Dougherty-Bliss and Manuel Kauers, Hardinian Arrays, arXiv:2309.00487 [math.CO], 2023.

Manuel Kauers and Christoph Koutschan, Guessing with Little Data, ISSAC '22: Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation, July 2022, Pages 83-90.

Manuel Kauers and Christoph Koutschan, Some D-finite and Some Possibly D-finite Sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023.

Formula

Recurrence: 32*(1 + n)*(1 + 2*n)^2*(161046 + 465785*n + 551943*n^2 + 343020*n^3 + 117954*n^4 + 21285*n^5 + 1575*n^6)*a(n) - 8*(4443102 + 33718283*n + 105734340*n^2 + 180574335*n^3 + 186866686*n^4 + 122556360*n^5 + 51280818*n^6 + 13267683*n^7 + 1933470*n^8 + 121275*n^9)*a(n+1) + 2*(12137328 + 91378536*n + 283626704*n^2 + 478464380*n^3 + 488415476*n^4 + 315713355*n^5 + 130145646*n^6 + 33170868*n^7 + 4763070*n^8 + 294525*n^9)*a(n+2) + (10688508 + 80866406*n + 252913504*n^2 + 431097970*n^3 + 445804136*n^4 + 292620525*n^5 + 122735586*n^6 + 31877118*n^7 + 4668570*n^8 + 294525*n^9)*a(n+3) - (4877748 + 36871922*n + 114948300*n^2 + 194784258*n^3 + 199650088*n^4 + 129484209*n^5 + 53503836*n^6 + 13655808*n^7 + 1961820*n^8 + 121275*n^9)*a(n+4) + 2*(3 + n)^2*(7 + 2*n)*(2428 + 16118*n + 41382*n^2 + 52554*n^3 + 35154*n^4 + 11835*n^5 + 1575*n^6)*a(n+5) = 0. - conjectured by Manuel Kauers and Christoph Koutschan, Mar 02 2023; proved by Robert Dougherty-Bliss and Manuel Kauers

Conjecture: a(n) ~ 2^(4*n - 2) / (81 * Pi * n), based on the above recurrence - Vaclav Kotesovec, Mar 02 2023

Example

Some solutions for n=4:
  0  1  1  1    0  1  1  1    0  0  0  1    0  0  0  1    0  0  0  1
  0  1  1  1    0  1  1  1    0  0  1  1    0  0  0  1    0  0  1  1
  0  1  1  1    1  1  1  1    0  1  1  1    0  0  1  1    0  0  1  1
  1  1  1  1    1  1  1  1    1  1  1  1    1  1  1  1    1  1  1  1

Crossrefs

Sequence in context: A036736 A016254 A016302 * A245237 A141942 A181043

Adjacent sequences: A253214 A253215 A253216 * A253218 A253219 A253220

Keyword

nonn

Author

R. H. Hardin, Dec 29 2014

Status

approved