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%I #28 Mar 10 2023 02:25:15
%S 1,1,16,985,141696,36372976,14083834704,7372392431849,
%T 4848332563899256,3808369342900073856,3447336241467721584256,
%U 3503140094024746011745456,3918646197894288330216058576,4753102567048482059557067412816,6178133154985813161258658378449616
%N Number of 5 X n matrices containing a permutation of 1..5*n in increasing order rowwise, columnwise, diagonally and (downwards) antidiagonally.
%H Christoph Koutschan, <a href="/A181199/b181199.txt">Table of n, a(n) for n = 1..100</a> (terms 1..26 from Alois P. Heinz)
%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, pp. 38-40.
%F Conjectured recurrence of order 3 and degree 24: 3*(n + 2)^2*(2*n + 3)*(2*n + 5)^2*(3*n + 7)*(3*n + 8)*(4*n + 9)*(4*n + 11)*(137855872*n^11 + 860969696*n^10 + 2047036856*n^9 + 2032587274*n^8 - 24192441*n^7 - 1894061166*n^6 - 1671661480*n^5 - 524330624*n^4 + 36004789*n^3 + 62751860*n^2 + 13865604*n + 927360)*(n + 3)^4*a(n + 3) - (n + 2)^2*(2*n + 3)*(717088749346816*n^21 + 14962008250398720*n^20 + 139665528127686656*n^19 + 760770273998231808*n^18 + 2616700630350746208*n^17 + 5556799141672247640*n^16 + 5462710847171622988*n^15 - 6080171043591548610*n^14 - 31771892184486994333*n^13 - 54791308745183340309*n^12 - 46634308960957698567*n^11 - 1215620047630796049*n^10 + 48833460779191449763*n^9 + 65692918122110754573*n^8 + 47097079083848116511*n^7 + 19628106098287909191*n^6 + 3499276436019940446*n^5 - 850259283025327524*n^4 - 685802053101717288*n^3 - 180089041888657440*n^2 - 22603977271411200*n - 1104708217536000)*a(n + 2) - 15*(2*n + 1)*(5*n + 6)*(5*n + 7)*(5*n + 8)*(5*n + 9)*(9113927409664*n^19 + 166164038596608*n^18 + 1357324488921088*n^17 + 6541981632511232*n^16 + 20586994844739808*n^15 + 44061606220301336*n^14 + 64302800289940820*n^13 + 61042725728660150*n^12 + 30740208891967721*n^11 - 3351043407516635*n^10 - 17410620603191989*n^9 - 12634296322883951*n^8 - 4487751704725767*n^7 - 793918042456325*n^6 - 54713722756179*n^5 + 42448206362469*n^4 + 50327965592874*n^3 + 21777752002716*n^2 + 4014999151560*n + 257403484800)*a(n + 1) + 25*(2*n - 1)^2*(2*n + 1)*(4*n - 1)*(4*n + 1)*(5*n + 1)*(5*n + 2)*(5*n + 3)*(5*n + 4)*(5*n + 6)*(5*n + 7)*(5*n + 8)*(5*n + 9)*(137855872*n^11 + 2377384288*n^10 + 18238806776*n^9 + 81945774178*n^8 + 238738633847*n^7 + 471293180347*n^6 + 638861237719*n^5 + 588363536007*n^4 + 354332674386*n^3 + 128320688700*n^2 + 23068181160*n + 1077753600)*a(n) = 0. - _Christoph Koutschan_, Feb 27 2023
%F Conjecture: a(n) ~ 9 * 5^(5*n + 1/2) / (2^17 * Pi^2 * n^12), based on the recurrence by _Christoph Koutschan_. - _Vaclav Kotesovec_, Feb 27 2023
%e Some solutions for 5 X 4:
%e 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
%e 5 6 7 8 5 6 7 8 5 6 7 8 5 6 7 8 5 6 7 8
%e 9 10 11 12 9 10 11 12 9 10 11 12 9 10 11 12 9 10 11 12
%e 13 14 15 16 13 14 15 17 13 14 15 18 13 14 16 17 13 14 16 18
%e 17 18 19 20 16 18 19 20 16 17 19 20 15 18 19 20 15 17 19 20
%t Table[
%t NextPartitions[n1_, n2_, n3_, n4_, n5_] :=
%t If[n1 < n, f[n1 + 1, n2, n3, n4, n5], 0] +
%t If[n2 < n1 - 1 || n2 === n - 1, f[n1, n2 + 1, n3, n4, n5], 0] +
%t If[n3 < n2 - 1 || n3 === n - 1 === n2 - 1, f[n1, n2, n3 + 1, n4, n5], 0] +
%t If[n4 < n3 - 1 || n4 === n - 1 === n3 - 1, f[n1, n2, n3, n4 + 1, n5], 0] +
%t If[n5 < n4 - 1, f[n1, n2, n3, n4, n5 + 1], 0];
%t pp = f[1, 0, 0, 0, 0];
%t Do[pp = Expand[pp /. f[ns__] :> NextPartitions[ns]], {5 n - 2}];
%t pp /. f[n, n, n, n, n - 1] -> 1,
%t {n, 20}] (* _Christoph Koutschan_, Feb 27 2023 *)
%Y Row 5 of A181196.
%K nonn
%O 1,3
%A _R. H. Hardin_, Oct 10 2010
%E a(11)-a(26) from _Alois P. Heinz_, Jul 24 2012