A390904 a(n) is the number of 3 X 3 matrices with elements 1..n where at least one row is strictly increasing.

0, 0, 0, 2107, 46144, 432250, 2548160, 11135495, 39398912, 119084364, 318528000, 772705395, 1730491840, 3625678342, 7179774784, 13546271375, 24507822080, 42739759160, 72155446272, 118351228779, 189171144000, 295414111090, 451709030080, 677586082327

Offset

0,4

Links

Table of n, a(n) for n=0..23.

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Formula

a(n) = n^9 - (n^3 * (1 + n)^3 * (5n - 2)^3)/216.

a(n) = n^9 - (n^3 - binomial(n, 3))^3.

G.f.: x^3*(2107 + 25074*x + 65625*x^2 + 49300*x^3 + 10335*x^4 + 438*x^5 + x^6)/(1 - x)^10. - Stefano Spezia, Nov 23 2025

Mathematica

a[n_, d_] := a[n, d] = AnyTrue[#, OrderedQ[#, Less] &] & /@ Tuples[Range[1, n], {d, d}] // Boole // Total;
Table[a[n, 3], {n, 0, 7}]

Crossrefs

Cf. A206808 (2 X 2 matrices).

Sequence in context: A233729 A272770 A373527 * A284962 A146895 A300008

Adjacent sequences: A390901 A390902 A390903 * A390905 A390906 A390907

Keyword

nonn,easy

Author

Robert P. P. McKone, Nov 23 2025

Status

approved