cp's OEIS Frontend

A055005 Number of nonnegative integer 3 X 3 matrices with no zero rows or columns and with sum of elements equal to n.

%I A055005 #13 Aug 14 2023 10:12:39
%S A055005 1,0,0,6,63,306,1038,2844,6750,14437,28521,52911,93258,157509,256581,
%T A055005 405171,622719,934542,1373158,1979820,2806281,3916812,5390496,7323822,
%U A055005 9833604,13060251,17171415,22366045,28878876,36985383,47007231
%N A055005 Number of nonnegative integer 3 X 3 matrices with no zero rows or columns and with sum of elements equal to n.
%H A055005 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F A055005 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 A055005 For p=q=3 we get a(n)=C(n + 8, 8) - 6*C(n + 5, 5) + 9*C(n + 3, 3) + 6*C(n + 2, 2) - 18*C(n + 1, 1) + 9=(1/8!)*(n^8 + 36*n^7 + 546*n^6 + 2520*n^5 - 7791*n^4 - 43596*n^3 + 148364*n^2 - 140400*n + 40320).
%F A055005 G.f.: -(9*x^8-54*x^7+132*x^6-171*x^5+135*x^4-78*x^3+36*x^2-9*x+1) / (x-1)^9. - _Colin Barker_, Jul 13 2013
%o A055005 (PARI) a(n)=([0,1,0,0,0,0,0,0,0; 0,0,1,0,0,0,0,0,0; 0,0,0,1,0,0,0,0,0; 0,0,0,0,1,0,0,0,0; 0,0,0,0,0,1,0,0,0; 0,0,0,0,0,0,1,0,0; 0,0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,0,1; 1,-9,36,-84,126,-126,84,-36,9]^n*[1;0;0;6;63;306;1038;2844;6750])[1,1] \\ _Charles R Greathouse IV_, Aug 14 2023
%Y A055005 Cf. A054688, A054974, A054975.
%K A055005 easy,nonn
%O A055005 0,4
%A A055005 _Vladeta Jovovic_, May 30 2000