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

Author

R. H. Hardin, Dec 29 2014

Keywords

• nonn

Comments

Diagonal of A253223.

Examples

			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
		

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