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A185784 Accumulation array of A107985, by antidiagonals. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Let W be the array given by w(1,1)=1, w(2,2)=-1, and w(n,k)=0 for all other (n,k).

Write "A < B" to indicate that an array B is the accumulation array of A (defined at A144112). Then  W < A103451 < A002024 < A107985 < A185784 < A185785 < A185786.

LINKS

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

FORMULA

T(n,k) = (n+k+1)*C(n+1,2)*C(k+1,2)/3, k>=0, n>=0.

EXAMPLE

Northwest corner:

1....4....10....20....35

4....15...36....70....120

10...36...84....160...270

20...70...160...300...500

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 *)

CROSSREFS

Cf. A144112, A103451, A002024, A107985, A185784, A185785, A185786.

Sequence in context: A180964 A237668 A209423 * A185904 A201618 A050339

Adjacent sequences:  A185781 A185782 A185783 * A185785 A185786 A185787

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 03 2011

STATUS

approved

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Last modified January 28 09:52 EST 2020. Contains 331319 sequences. (Running on oeis4.)