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

1, 4, 4, 10, 16, 10, 20, 40, 40, 20, 35, 80, 100, 80, 35, 56, 140, 200, 200, 140, 56, 84, 224, 350, 400, 350, 224, 84, 120, 336, 560, 700, 700, 560, 336, 120, 165, 480, 840, 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

Offset: 1

Author

Clark Kimberling, Feb 06 2011

Keywords

• nonn
• tabl

Comments

A member of the accumulation chain ... < A185906 < A000007 < A003991 < A098358 < A185904 < A185905 < ... (See A144112 for the definition of accumulation array.)

Examples

			Northwest corner:
   1,  4,  10,  20,  35
   4, 16,  40,  80, 140
  10, 40, 100, 200, 350
  20, 80, 200, 400, 700
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Cf. A000007, A003991, A098358, A144112, A185905, A185906, A185907.

Row 1 = Column 1 = A000292.

Programs

Mathematica

(* This program generates A098358 and its accumulation array, A185904. *)
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A098358 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
FullSimplify[s[n,k]]  (* formula for A185904 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A185904 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
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 *)

Formula

T(n,k) = binomial(k+2,3)*binomial(n+2,3), k >= 1, n >= 1.