A047922 Triangle of numbers a(n,k) = number of terms in n X n determinant with 2 adjacent diagonals of k and k-1 0's (0<=k<=n).

1, 1, 0, 2, 1, 0, 6, 4, 1, 1, 24, 18, 8, 5, 3, 120, 96, 54, 34, 23, 16, 720, 600, 384, 258, 182, 131, 96, 5040, 4320, 3000, 2136, 1566, 1168, 883, 675, 40320, 35280, 25920, 19320, 14664, 11274, 8756, 6859, 5413, 362880, 322560, 246960, 190800, 149160, 117696, 93582, 74902, 60301, 48800

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• nice
• easy

Examples

			Triangle starts:
  1;
  1, 0;
  2, 1, 0;
  6, 4, 1, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• J. D. H. Dickson, Discussion of two double series arising from the number of terms in determinants of certain forms, Proc. London Math. Soc., 10 (1879), 120-122.
• J. D. H. Dickson, Discussion of two double series arising from the number of terms in determinants of certain forms, Proc. London Math. Soc., 10 (1879), 120-122. [Annotated scanned copy]

Crossrefs

Columns give A000142, A001563, A002775, A002776. Cf. A047920.

Programs

Maple

a:= proc(n, k) option remember; `if`(k=0, n!, `if`(n=k,
      `if`(n<3, (n-1)*(n-2)/2, (n-1)*(a(n-1$2)+a(n-2$2))
      +a(n-3$2)), a(n, k+1) +2*a(n-1, k) +a(n-2, k-1)))
    end:
seq(seq(a(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jun 24 2017

Mathematica

a[n_, n_] := (-1)^n*HypergeometricPFQ[{1, -n, n+1}, {1/2}, 1/4]; a[n_, k_] := a[n, k] = a[n, k+1] + 2*a[n-1, k] + a[n-2, k-1]; Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 24 2015 *)

Formula

Right diagonal is A000271, column k=0 is A000142; other entries given by a(n, k) = a(n, k+1) + 2a(n-1, k) + a(n-2, k-1).

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Sep 29 2000