cp's OEIS Frontend

A055007 Number of nonnegative integer 4 X 4 matrices with no zero rows or columns and with sum of elements equal to n.

%I A055007 #18 Jul 02 2025 16:01:59
%S A055007 1,0,0,0,24,528,4648,26224,112666,401424,1246000,3476368,8905432,
%T A055007 21266208,47875272,102482048,210000931,414160240,789572072,1460372624,
%U A055007 2628456428,4615495808,7924479264,13328517504,21997272036,35674700896,56926058920,89477437120
%N A055007 Number of nonnegative integer 4 X 4 matrices with no zero rows or columns and with sum of elements equal to n.
%H A055007 T. D. Noe, <a href="/A055007/b055007.txt">Table of n, a(n) for n = 0..1000</a>
%H A055007 <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).
%F A055007 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).
%F A055007 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).
%F A055007 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
%Y A055007 Cf. A054688, A054974, A054975, A052366.
%K A055007 easy,nonn
%O A055007 0,5
%A A055007 _Vladeta Jovovic_, May 30 2000
%E A055007 More terms from _James Sellers_, May 31 2000