A185784 Accumulation array of A107985, by antidiagonals.

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

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: A366754 A209423 A357620 * A185904 A201618 A341243

Adjacent sequences: A185781 A185782 A185783 * A185785 A185786 A185787

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 03 2011

Status

approved