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A185904 Multiplication table for the tetrahedral numbers (A000292), by antidiagonals.

%I A185904 #14 Jul 22 2017 09:14:18
%S A185904 1,4,4,10,16,10,20,40,40,20,35,80,100,80,35,56,140,200,200,140,56,84,
%T A185904 224,350,400,350,224,84,120,336,560,700,700,560,336,120,165,480,840,
%U A185904 1120,1225,1120,840,480,165,220,660,1200,1680,1960,1960,1680,1200,660,220,286,880,1650,2400,2940,3136,2940,2400,1650,880,286,364,1144,2200,3300,4200,4704,4704,4200,3300,2200,1144,364,455,1456,2860,4400,5775,6720,7056,6720,5775,4400,2860,1456,455,560,1820,3640,5720,7700,9240,10080,10080,9240,7700,5720,3640,1820,560
%N A185904 Multiplication table for the tetrahedral numbers (A000292), by antidiagonals.
%C A185904 A member of the accumulation chain ... < A185906 < A000007 < A003991 < A098358 < A185904 < A185905 < ...  (See A144112 for the definition of accumulation array.)
%H A185904 G. C. Greubel, <a href="/A185904/b185904.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A185904 T(n,k) = binomial(k+2,3)*binomial(n+2,3), k >= 1, n >= 1.
%e A185904 Northwest corner:
%e A185904    1,  4,  10,  20,  35
%e A185904    4, 16,  40,  80, 140
%e A185904   10, 40, 100, 200, 350
%e A185904   20, 80, 200, 400, 700
%t A185904 (* This program generates A098358 and its accumulation array, A185904. *)
%t A185904 TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A098358 *)
%t A185904 Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%t A185904 s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
%t A185904 FullSimplify[s[n,k]]  (* formula for A185904 *)
%t A185904 TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A185904 *)
%t A185904 Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%t A185904 T[n_, k_] := Binomial[k + 2, 3]*Binomial[n + 2, 3]; Table[T[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* _G. C. Greubel_, Jul 22 2017 *)
%Y A185904 Cf. A000007, A003991, A098358, A144112, A185905, A185906, A185907.
%Y A185904 Row 1 = Column 1 = A000292.
%K A185904 nonn,tabl
%O A185904 1,2
%A A185904 _Clark Kimberling_, Feb 06 2011