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A212250 Number of (w,x,y,z) with all terms in {1,...,n} and 3w>=x+y+z+n.

%I A212250 #15 Jan 05 2025 12:31:26
%S A212250 0,0,4,18,54,129,262,478,807,1281,1938,2821,3975,5451,7305,9595,12385,
%T A212250 15744,19743,24459,29974,36372,43743,52182,61786,72658,84906,98640,
%U A212250 113976,131035,149940,170820,193809,219043,246664,276819,309657
%N A212250 Number of (w,x,y,z) with all terms in {1,...,n} and 3w>=x+y+z+n.
%C A212250 a(n)+A212249 = n^4.
%C A212250 For a guide to related sequences, see A211795.
%H A212250 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,5,-5,6,-4,1).
%F A212250 a(n) = 4*a(n-1)-6*a(n-2)+5*a(n-3)-5*a(n-4)+6*a(n-5)-4*a(n-6)+a(n-7).
%F A212250 G.f.: x^2*(4+2*x+6*x^2+x^3)/((1+x+x^2)*(1-x)^5). - _Bruno Berselli_, Jun 05 2012
%F A212250 a(n) = (13*n^4+10*n^3-5*n^2+6*n+8*b)/72, where b = 0,-3,1,0,-3,1,... (repeated). [_Bruno Berselli_, Jun 05 2012]
%t A212250 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212250 (Do[If[3 w >= x + y + z + n, s = s + 1],
%t A212250 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212250 Map[t[#] &, Range[0, 40]]  (* A212250 *)
%t A212250 (* _Peter J. C. Moses_, Apr 13 2012 *)
%Y A212250 Cf. A211795, A212247, A212249.
%K A212250 nonn,easy
%O A212250 0,3
%A A212250 _Clark Kimberling_, May 09 2012