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%I #31 Apr 04 2024 10:56:21
%S 0,0,1,19,268,3568,47698,649712,9023385,127419681,1823918697,
%T 26398702645,385582981615,5674890516295,84060883775765,
%U 1252066289632643,18738613233957420,281620474177057788,4248088188086420832
%N 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.
%C Diagonal of A253223.
%H R. H. Hardin, <a href="/A253217/b253217.txt">Table of n, a(n) for n = 1..37</a>
%H Robert Dougherty-Bliss, <a href="https://sites.math.rutgers.edu/~zeilberg/Theses/RobertDoughertyBlissThesis.pdf">Experimental Methods in Number Theory and Combinatorics</a>, Ph. D. Dissertation, Rutgers Univ. (2024). See p. 29.
%H Robert Dougherty-Bliss and Manuel Kauers, <a href="https://arxiv.org/abs/2309.00487">Hardinian Arrays</a>, arXiv:2309.00487 [math.CO], 2023.
%H Manuel Kauers and Christoph Koutschan, <a href="https://doi.org/10.1145/3476446.3535486">Guessing with Little Data</a>, ISSAC '22: Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation, July 2022, Pages 83-90.
%H Manuel Kauers and Christoph Koutschan, <a href="https://arxiv.org/abs/2303.02793">Some D-finite and Some Possibly D-finite Sequences in the OEIS</a>, arXiv:2303.02793 [cs.SC], 2023.
%F 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_
%F Conjecture: a(n) ~ 2^(4*n - 2) / (81 * Pi * n), based on the above recurrence - _Vaclav Kotesovec_, Mar 02 2023
%e Some solutions for n=4:
%e 0 1 1 1 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1
%e 0 1 1 1 0 1 1 1 0 0 1 1 0 0 0 1 0 0 1 1
%e 0 1 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 0 1 1
%e 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%K nonn
%O 1,4
%A _R. H. Hardin_, Dec 29 2014