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

1,2

Comments

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

Links

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

Formula

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

Example

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

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 *)

Crossrefs

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

Row 1 = Column 1 = A000292.

Sequence in context: A209423 A357620 A185784 * A201618 A341243 A050339

Adjacent sequences: A185901 A185902 A185903 * A185905 A185906 A185907

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 06 2011

Status

approved