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A185904 Multiplication table for the tetrahedral numbers (A000292), by antidiagonals. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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: A237668 A209423 A185784 * A201618 A050339 A087288

Adjacent sequences:  A185901 A185902 A185903 * A185905 A185906 A185907

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 06 2011

STATUS

approved

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Last modified October 19 20:05 EDT 2018. Contains 316378 sequences. (Running on oeis4.)