A181199 Number of 5 X n matrices containing a permutation of 1..5*n in increasing order rowwise, columnwise, diagonally and (downwards) antidiagonally.

1, 1, 16, 985, 141696, 36372976, 14083834704, 7372392431849, 4848332563899256, 3808369342900073856, 3447336241467721584256, 3503140094024746011745456, 3918646197894288330216058576, 4753102567048482059557067412816, 6178133154985813161258658378449616

Offset

1,3

Links

Christoph Koutschan, Table of n, a(n) for n = 1..100 (terms 1..26 from Alois P. Heinz)

Manuel Kauers and Christoph Koutschan, Some D-finite and some Possibly D-finite Sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023, pp. 38-40.

Formula

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

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

Example

Some solutions for 5 X 4:
  1  2  3  4    1  2  3  4    1  2  3  4    1  2  3  4    1  2  3  4
  5  6  7  8    5  6  7  8    5  6  7  8    5  6  7  8    5  6  7  8
  9 10 11 12    9 10 11 12    9 10 11 12    9 10 11 12    9 10 11 12
 13 14 15 16   13 14 15 17   13 14 15 18   13 14 16 17   13 14 16 18
 17 18 19 20   16 18 19 20   16 17 19 20   15 18 19 20   15 17 19 20

Mathematica

Table[
  NextPartitions[n1_, n2_, n3_, n4_, n5_] :=
    If[n1 < n, f[n1 + 1, n2, n3, n4, n5], 0] +
    If[n2 < n1 - 1 || n2 === n - 1, f[n1, n2 + 1, n3, n4, n5], 0] +
    If[n3 < n2 - 1 || n3 === n - 1 === n2 - 1, f[n1, n2, n3 + 1, n4, n5], 0] +
    If[n4 < n3 - 1 || n4 === n - 1 === n3 - 1, f[n1, n2, n3, n4 + 1, n5], 0] +
    If[n5 < n4 - 1, f[n1, n2, n3, n4, n5 + 1], 0];
  pp = f[1, 0, 0, 0, 0];
  Do[pp = Expand[pp /. f[ns__] :> NextPartitions[ns]], {5 n - 2}];
  pp /. f[n, n, n, n, n - 1] -> 1,
{n, 20}] (* Christoph Koutschan, Feb 27 2023 *)

Crossrefs

Row 5 of A181196.

Sequence in context: A264640 A211086 A187802 * A024301 A211090 A160012

Adjacent sequences: A181196 A181197 A181198 * A181200 A181201 A181202

Keyword

nonn

Author

R. H. Hardin, Oct 10 2010

Extensions

a(11)-a(26) from Alois P. Heinz, Jul 24 2012

Status

approved