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A212560 Number of (w,x,y,z) with all terms in {1,...,n} and w+x<=y+z.

%I A212560 #25 Oct 21 2022 21:06:46
%S A212560 0,1,11,50,150,355,721,1316,2220,3525,5335,7766,10946,15015,20125,
%T A212560 26440,34136,43401,54435,67450,82670,100331,120681,143980,170500,
%U A212560 200525,234351,272286,314650,361775,414005,471696,535216,604945,681275
%N A212560 Number of (w,x,y,z) with all terms in {1,...,n} and w+x<=y+z.
%C A212560 For a guide to related sequences, see A211795.
%C A212560 For n>=1, a(n) is the n-th antidiagonal sums of the convolution array A213831.  - _Clark Kimberling_, Jul 04 2012
%H A212560 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A212560 a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5).
%F A212560 a(n) = (n + 2*n^3 + 3*n^4)/6. - _Clark Kimberling_, Jul 10 2012
%F A212560 G.f.: x*(1 + x)*(1 + 5*x)/(1 - x)^5. - _Clark Kimberling_, Jul 10 2012
%F A212560 a(n) = Sum_{k=0..n} A059722(k). - _J. M. Bergot_, Nov 02 2012
%t A212560 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212560 (Do[If[w + x <= y + z, s = s + 1],
%t A212560 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212560 Map[t[#] &, Range[0, 40]]   (* A212560 *)
%o A212560 (PARI) a(n)=(n+2*n^3+3*n^4)/6 \\ _Charles R Greathouse IV_, Oct 21 2022
%Y A212560 Cf. A211795.
%K A212560 nonn,easy
%O A212560 0,3
%A A212560 _Clark Kimberling_, May 21 2012