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

0, 1, 11, 50, 150, 355, 721, 1316, 2220, 3525, 5335, 7766, 10946, 15015, 20125, 26440, 34136, 43401, 54435, 67450, 82670, 100331, 120681, 143980, 170500, 200525, 234351, 272286, 314650, 361775, 414005, 471696, 535216, 604945, 681275

Offset: 0

Author

Clark Kimberling, May 21 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211795.

For n>=1, a(n) is the n-th antidiagonal sums of the convolution array A213831. - Clark Kimberling, Jul 04 2012

Links

• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A211795.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w + x <= y + z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212560 *)

PARI

a(n)=(n+2*n^3+3*n^4)/6 \\ Charles R Greathouse IV, Oct 21 2022

Formula

a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5).

a(n) = (n + 2*n^3 + 3*n^4)/6. - Clark Kimberling, Jul 10 2012

G.f.: x*(1 + x)*(1 + 5*x)/(1 - x)^5. - Clark Kimberling, Jul 10 2012

a(n) = Sum_{k=0..n} A059722(k). - J. M. Bergot, Nov 02 2012