A058389 Number of 3 X 3 matrices with nonnegative integer entries and all row sums equal to n, up to row and column permutation.

1, 3, 14, 44, 129, 316, 714, 1452, 2775, 4963, 8478, 13838, 21827, 33306, 49504, 71754, 101871, 141807, 194128, 261570, 347633, 456026, 591384, 758596, 963657, 1212861, 1513806, 1874440, 2304225, 2813030, 3412466, 4114608, 4933519

Offset

0,2

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

V. Jovovic, Illustration of initial terms

V. Jovovic, Number of m x m nonnegative integer matrices with all row sums equal to n, up to row and column permutation.

Formula

a(n) = (1/6)*(C(C(n + 2, 2) + 2, 3) + 3/2*floor((n + 2)/2)*(C(n + 2, 2) - floor((n + 2)/2)) + 3*C(floor((n + 2)/2) + 2, 3) + 2*floor(C(n + 2, 2)/3) + 2*C(C(n + 2, 2) - 3*floor(C(n + 2, 2)/3) + 2, 3)).

Empirical G.f.: -(x^8 + 3*x^7 + 14*x^6 + 12*x^5 + 15*x^4 + 9*x^3 + 5*x^2 + 1) / ((x-1)^7*(x+1)^3*(x^2+x+1)). - Colin Barker, Dec 27 2012

Mathematica

a[n_] := (m = Mod[n, 6]; (n^3 + 9*n^2 + 39*n + 120)*n^3 + Which[m == 0, 12*(23*n^2 + 32*n + 24), m == 1 || m == 5, 249*n^2 + 303*n + 143, m == 2 || m == 4, 4*(69*n^2 + 96*n + 56), m == 3, 3*(83*n^2 + 101*n + 69)])/288; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Oct 12 2011, after Vladeta Jovovic *)

Prog

(PARI) \\ See A318951 for RowSumMats
a(n)=RowSumMats(3, 3, n); \\ Andrew Howroyd, Sep 05 2018

Crossrefs

Row 3 of A318951.

Cf. A002817, A052282, A058390, A058391, A058392.

Cf. A001501, A050535, A050913, A058528, A058783, A058784, A058785.

Sequence in context: A063903 A305009 A115005 * A261481 A059672 A302225

Adjacent sequences: A058386 A058387 A058388 * A058390 A058391 A058392

Keyword

nice,nonn,easy

Author

Vladeta Jovovic, Nov 24 2000

Extensions

More terms from Marc LeBrun, Dec 11 2000

Status

approved