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A210377 Number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = n + 5.

%I A210377 #11 Jul 16 2024 13:17:34
%S A210377 0,0,4,31,80,146,224,315,420,540,676,829,1000,1190,1400,1631,1884,
%T A210377 2160,2460,2785,3136,3514,3920,4355,4820,5316,5844,6405,7000,7630,
%U A210377 8296,8999,9740,10520,11340,12201,13104,14050,15040,16075,17156,18284
%N A210377 Number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = n + 5.
%C A210377 A210376 is also the number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = 3n - 5.
%C A210377 See A210000 for a guide to related sequences.
%H A210377 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4, -6, 4, -1).
%F A210377 Conjectures from _Colin Barker_, Dec 07 2017: (Start)
%F A210377 G.f.: x^2*(4 + 15*x - 20*x^2 - 4*x^3 + 6*x^5) / (1 - x)^4.
%F A210377 a(n) = (-504 + 146*n + 21*n^2 + n^3) / 6 for n>3.
%F A210377 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>5.
%F A210377 (End)
%t A210377 a = 0; b = n; z1 = 45;
%t A210377 t[n_] := t[n] = Flatten[Table[w + x + y + z, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
%t A210377 c[n_, k_] := c[n, k] = Count[t[n], k]
%t A210377 Table[c[n, n + 5], {n, 0, z1}]    (* A210377 *)
%t A210377 Table[c[n, 3 n - 5], {n, 0, z1}]  (* A210377 *)
%Y A210377 Cf. A210000.
%K A210377 nonn
%O A210377 0,3
%A A210377 _Clark Kimberling_, Mar 20 2012