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

Author

R. H. Hardin, Oct 10 2010

Keywords

• nonn

Examples

			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
		

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.

Crossrefs

Row 4 of A181196.

Programs

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 *)

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.

Extensions

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

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