A055007 Number of nonnegative integer 4 X 4 matrices with no zero rows or columns and with sum of elements equal to n.
1, 0, 0, 0, 24, 528, 4648, 26224, 112666, 401424, 1246000, 3476368, 8905432, 21266208, 47875272, 102482048, 210000931, 414160240, 789572072, 1460372624, 2628456428, 4615495808, 7924479264, 13328517504, 21997272036, 35674700896, 56926058920, 89477437120
Offset: 0
Links
- T. D. Noe, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).
Formula
Number of nonnegative integer p X q matrices with no zero rows or columns and with sum of elements equal to n is Sum_{k=0...q} (-1)^k*C(q, k)*m(p, q-k, n) where m(p, q, n)=Sum_{k=0..p} (-1)^k*C(p, k)*C((p-k)*q+n-1, n).
For p = q = 4 we get a(n) = (1/15!)*(n^15 + 120*n^14 + 6580*n^13 + 218400*n^12 + 4637542*n^11 + 61261200*n^10 + 423591740*n^9 + 164392800*n^8 - 17247717487*n^7 - 47940252360*n^6 + 346941238280*n^5 + 557885764800*n^4 - 4897231459056*n^3 + 8643549191040*n^2 - 5894285241600*n + 1307674368000).
G.f.: -(16*x^15 -192*x^14 +1040*x^13 -3356*x^12 +7200*x^11 -10952*x^10 +12544*x^9 -11712*x^8 +9664*x^7 -7088*x^6 +4224*x^5 -1844*x^4 +560*x^3 -120*x^2 +16*x -1) / (x -1)^16. - Colin Barker, Jul 11 2013
Extensions
More terms from James Sellers, May 31 2000