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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).
5
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
0,4
LINKS
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]
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).
EXAMPLE
Triangle starts:
1;
1, 0;
2, 1, 0;
6, 4, 1, 1;
...
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 *)
CROSSREFS
Columns give A000142, A001563, A002775, A002776. Cf. A047920.
Sequence in context: A122538 A090238 A358694 * A276891 A021830 A247686
KEYWORD
nonn,tabl,nice,easy
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Sep 29 2000
STATUS
approved