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A210374 Number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = n+2.

%I A210374 #11 Jul 16 2024 13:11:50
%S A210374 0,4,19,40,68,104,149,204,270,348,439,544,664,800,953,1124,1314,1524,
%T A210374 1755,2008,2284,2584,2909,3260,3638,4044,4479,4944,5440,5968,6529,
%U A210374 7124,7754,8420,9123,9864,10644,11464,12325,13228,14174,15164
%N A210374 Number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = n+2.
%C A210374 A210374 is also the number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = 3n-2.
%C A210374 See A210000 for a guide to related sequences.
%H A210374 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4, -6, 4, -1).
%F A210374 Conjectures from _Colin Barker_, Dec 07 2017: (Start)
%F A210374 G.f.: x*(4 + 3*x - 12*x^2 + 6*x^3) / (1 - x )^4.
%F A210374 a(n) = (-36 + 47*n + 12*n^2 + n^3) / 6 for n>0.
%F A210374 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.
%F A210374 (End)
%t A210374 a = 0; b = n; z1 = 45;
%t A210374 t[n_] := t[n] = Flatten[Table[w + x + y + z, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
%t A210374 c[n_, k_] := c[n, k] = Count[t[n], k]
%t A210374 Table[c[n, n + 2], {n, 0, z1}]    (* A210374 *)
%t A210374 Table[c[n, 3 n - 2], {n, 0, z1}]  (* A210374 *)
%Y A210374 Cf. A210000.
%K A210374 nonn
%O A210374 0,2
%A A210374 _Clark Kimberling_, Mar 20 2012