A181198 Number of 4 X n matrices containing a permutation of 1..4*n in increasing order rowwise, columnwise, diagonally and (downwards) antidiagonally.

1, 1, 8, 169, 6392, 352184, 25097600, 2152061145, 212012802584, 23263015359672, 2781709560836960, 356806123331844056, 48516442013911012288, 6930091952294051922080, 1032505514388962439665280, 159544871422153344631037625, 25451354639006231998529405016

Offset

1,3

Links

Christoph Koutschan, Table of n, a(n) for n = 1..100 (terms 1..27 from Alois P. Heinz)

Joerg Arndt, The a(4)=199 shifted standard Young tableaux of shape [4,4,4,4]

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 2 and degree 9: 3*(n + 1)*(2*n + 3)*(3*n + 4)*(3*n + 5)*(7*n^2 - 1)*(n + 2)^3*a(n + 2) - 8*(n + 1)*(2*n + 1)*(4*n + 3)*(4*n + 5)*(364*n^5 + 84*n^4 - 1025*n^3 - 534*n^2 + 157*n + 54)*a(n + 1) - 64*(2*n - 1)^2*(2*n + 1)*(4*n - 1)*(4*n + 1)*(4*n + 3)*(4*n + 5)*(7*n^2 + 14*n + 6)*a(n) = 0. - Christoph Koutschan, Feb 26 2023

Conjectured formula, solution to the above recurrence, for n > 1: a(n) = (-64)^n * (n-1) * (-1/2)_{2*n} * (1/2)_{n} / (4*(3*n)!) * (-1 + 3*Sum_{k=2..n-1} (-4)^k * (7*k^2-1) / ((k-1) * k * (k+1)^2 * (2*k-1)^2 * (2*k+1)^3) * binomial(3*k,2*k) * binomial(k+1/2,k)), where (a)_{n} is the Pochhammer symbol.

Example

Some solutions for 4 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 13    9 10 11 14    9 10 12 13    9 10 12 14
  13 14 15 16   12 14 15 16   12 13 15 16   11 14 15 16   11 13 15 16

Mathematica

Table[
 NextPartitions[n1_, n2_, n3_, n4_] :=
   If[n1 < n, f[n1 + 1, n2, n3, n4], 0] +
   If[n2 < n1 - 1 || n2 === n - 1, f[n1, n2 + 1, n3, n4], 0] +
   If[n3 < n2 - 1 || n3 === n - 1 === n2 - 1, f[n1, n2, n3 + 1, n4], 0] +
   If[n4 < n3 - 1, f[n1, n2, n3, n4 + 1], 0];
 pp = f[1, 0, 0, 0];
 Do[pp = Expand[pp /. f[ns__] :> NextPartitions[ns]], {4 n - 2}];
 pp /. f[n, n, n, n - 1] -> 1,
{n, 20}] (* Christoph Koutschan, Feb 26 2023 *)

Crossrefs

Row 4 of A181196.

Sequence in context: A084941 A139564 A264113 * A302802 A316816 A001534

Adjacent sequences: A181195 A181196 A181197 * A181199 A181200 A181201

Keyword

nonn

Author

R. H. Hardin, Oct 10 2010

Extensions

a(12)-a(27) from Alois P. Heinz, Jul 24 2012

a(28)-a(100) from Christoph Koutschan, Feb 26 2023

Status

approved