A185784 Accumulation array of A107985, by antidiagonals.
1, 4, 4, 10, 15, 10, 20, 36, 36, 20, 35, 70, 84, 70, 35, 56, 120, 160, 160, 120, 56, 84, 189, 270, 300, 270, 189, 84, 120, 280, 420, 500, 500, 420, 280, 120, 165, 396, 616, 770, 825, 770, 616, 396, 165, 220, 540, 864, 1120, 1260, 1260, 1120, 864, 540, 220, 286, 715, 1170, 1560, 1820, 1911, 1820, 1560, 1170, 715, 286, 364, 924, 1540, 2100, 2520, 2744, 2744, 2520, 2100, 1540, 924, 364, 455, 1170, 1980, 2750, 3375, 3780, 3920, 3780, 3375, 2750, 1980, 1170, 455, 560
Offset: 1
Examples
Northwest corner: 1....4....10....20....35 4....15...36....70....120 10...36...84....160...270 20...70...160...300...500
Links
- G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Programs
-
Mathematica
(* The code generates arrays A107985, A185784, A002024. *) f[n_,0]:=0;f[0,k_]:=0; (* used to form A002024 *) f[n_,k_]:=k*n(k+n)/2; TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A107985 *) s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *) FullSimplify[s[n,k]] TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A185784 *) Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten w[m_,n_]:=f[m,n]+f[m-1,n-1]-f[m,n-1]-f[m-1,n]/;Or[m>0,n>0]; TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* A002024 *)
Formula
T(n,k) = (n+k+1)*C(n+1,2)*C(k+1,2)/3, k>=0, n>=0.
Comments