A185784 -id:A185784 - cp's OEIS frontend

A185785 Second accumulation array of A107985, by antidiagonals.

1, 5, 5, 15, 24, 15, 35, 70, 70, 35, 70, 160, 200, 160, 70, 126, 315, 450, 450, 315, 126, 210, 560, 875, 1000, 875, 560, 210, 330, 924, 1540, 1925, 1925, 1540, 924, 330, 495, 1440, 2520, 3360, 3675, 3360, 2520, 1440, 495, 715, 2145, 3900, 5460, 6370, 6370, 5460, 3900, 2145, 715, 1001, 3080, 5775, 8400, 10290, 10976, 10290, 8400, 5775, 3080, 1001

Offset: 1

Clark Kimberling, Feb 03 2011

See A185784.

			Northwest corner:
1....5....15....35....70
5....24...70....160...315
15...70...200...450...875
35...160..450...1000..1925

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

Cf. A185784.

f[n_, k_] := Binomial[k + 2, 3]*Binomial[n + 2, 3]*(n + k + 2)/4;
Table[f[n - k + 1, k], {n, 50}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 12 2017 *)

T(n,k) = (n+k+2)*C(n+2,3)*C(k+2,3)/4, k>=1, n>=1.

A185786 Third accumulation array of A107985, by antidiagonals.

Original entry on oeis.org

1, 6, 6, 21, 35, 21, 56, 120, 120, 56, 126, 315, 405, 315, 126, 252, 700, 1050, 1050, 700, 252, 462, 1386, 2310, 2695, 2310, 1386, 462, 792, 2520, 4536, 5880, 5880, 4536, 2520, 792, 1287, 4290, 8190, 11466, 12740, 11466, 8190, 4290, 1287, 2002, 6930, 13860, 20580, 24696, 24696, 20580, 13860, 6930, 2002, 3003, 10725, 22275, 34650

Offset: 1

Clark Kimberling, Feb 03 2011

See A185784. The pattern established by the formulas for A185785, A185786, A185787, suggests that the H-th accumulation array of A107985 may be given by

T(n,k)=(n+k+H)C(n+H,H+1)C(k+H,H+1)/(H+2).

			Northwest corner:
1....6.....21.....56.....126
6....35....120....315....700
21...120...405....1050...2310
56...315...1050...2695...5880

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

Cf. A185784.

(See A185784.)
f[n_, k_] := Binomial[k + 3, 4]*Binomial[n + 3, 4]*(n + k + 3)/5; Table[f[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 12 2017 *)

T(n,k) = (n+k+3)*C(n+3,4)*C(k+3,4)/5.

cp's OEIS Frontend

A185785 Second accumulation array of A107985, by antidiagonals.

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