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A212090 Number of (w,x,y,z) with all terms in {1,...,n} and w

%I A212090 #15 Dec 04 2016 19:46:29
%S A212090 0,1,16,80,251,610,1261,2331,3970,6351,9670,14146,20021,27560,37051,
%T A212090 48805,63156,80461,101100,125476,154015,187166,225401,269215,319126,
%U A212090 375675,439426,510966,590905,679876,778535,887561,1007656,1139545
%N A212090 Number of (w,x,y,z) with all terms in {1,...,n} and w<x+y+z.
%C A212090 a(n)+A000332(n+1) = n^4. For a guide to related sequences, see A211795.
%H A212090 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5, -10, 10, -5, 1).
%F A212090 a(n) = 5a(n-1)-10a(n-2)+10a(n-3)-5a(n-4)+a(n-5).
%F A212090 G.f.: -x*(1+11*x+10*x^2+x^3) / (x-1)^5.
%F A212090 a(n) = n*(-2+(1+(2+23*n)*n)*n)/24.
%t A212090 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212090 (Do[If[w < x + y + z, s = s + 1],
%t A212090 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212090 Map[t[#] &, Range[0, 50]] (* A212090 *)
%t A212090 FindLinearRecurrence[%]
%t A212090 (* _Peter J. C. Moses_, Apr 13 2012 *)
%t A212090 LinearRecurrence[{5, -10, 10, -5, 1},{0, 1, 16, 80, 251},34] (* _Ray Chandler_, Aug 02 2015 *)
%Y A212090 Cf. A211795, A212091.
%K A212090 nonn,easy
%O A212090 0,3
%A A212090 _Clark Kimberling_, May 01 2012