A185877 -id:A185877 - cp's OEIS frontend

A185878 Accumulation array of A185877, by antidiagonals.

1, 4, 2, 11, 10, 3, 24, 28, 18, 4, 45, 60, 51, 28, 5, 76, 110, 108, 80, 40, 6, 119, 182, 195, 168, 115, 54, 7, 176, 280, 318, 300, 240, 156, 70, 8, 249, 408, 483, 484, 425, 324, 203, 88, 9, 340, 570, 696, 728, 680, 570, 420, 256, 108, 10, 451, 770, 963, 1040, 1015, 906, 735, 528, 315, 130, 11, 584, 1012, 1290, 1428, 1440, 1344, 1162, 920, 648, 380, 154, 12

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain ... < A185879 < A185877 < A185878 < A185880 < ...

See A144112 for the definition of accumulation array.

			Northwest corner:
  1,  4, 11,  24,  45, ...
  2, 10, 28,  60, 110, ...
  3, 18, 51, 108, 195, ...
  4, 28, 80, 168, 300, ...
  ...

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

Cf. A144112, A185877, A185879.
Row 1 to 3: A006527, A006331, A064043.
Column 1 to 5: A000027, A028552, A140677, 12*A000096, 5*A130861.

f[n_, k_] := k*n*(2*k^2 - 3*k + 3*k*n - 3*n + 7)/6; Table[f[n - k + 1, k], {n,10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 21 2017 *)

T(n,k) = k*n*(2*k^2 -3*k +3*k*n -3*n +7)/6, k>=1, n>=1.

A185879 Weight array of A185877, by antidiagonals.

Original entry on oeis.org

1, 2, 0, 4, 2, 0, 6, 2, 2, 0, 8, 2, 2, 2, 0, 10, 2, 2, 2, 2, 0, 12, 2, 2, 2, 2, 2, 0, 14, 2, 2, 2, 2, 2, 2, 0, 16, 2, 2, 2, 2, 2, 2, 2, 0, 18, 2, 2, 2, 2, 2, 2, 2, 2, 0, 20, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 22, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 24, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 26, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain ...< A185879 < A185877 < A185878 < A185880 <...

See A144112 for the definitions of weight array and accumulation array.

			Northwest corner:
1...2...4...6...8...10...12...14
0...2...2...2...2...2....2....2
0...2...2...2...2...2....2....2
0...2...2...2...2...2....2....2
0...2...2...2...2...2....2....2

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

Cf. A144112, A185877.

f[n_, k_] := 2; f[1, k_] := 2*(k - 1); f[n_, 1] := 0; f[1, 1] := 1;
TableForm[Table[f[n, k], {n, 1, 7}, {k, 1, 7}]] Table[f[n - k + 1, k], {n, 5}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 21 2017 *)

T(1,k) = 2*(k-1) for k>=1; T(n,1)=0 for n>1; T(n,k)=2 otherwise.

A185880 Second accumulation array of A185877, by antidiagonals.

Original entry on oeis.org

1, 5, 3, 16, 17, 6, 40, 56, 38, 10, 85, 140, 128, 70, 15, 161, 295, 320, 240, 115, 21, 280, 553, 670, 600, 400, 175, 28, 456, 952, 1246, 1250, 1000, 616, 252, 36, 705, 1536, 2128, 2310, 2075, 1540, 896, 348, 45, 1045, 2355, 3408, 3920, 3815, 3185, 2240, 1248, 465, 55, 1496, 3465, 5190, 6240, 6440, 5831, 4620, 3120, 1680, 605, 66, 2080, 4928, 7590, 9450, 10200, 9800, 8428, 6420, 4200, 2200

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain ... < A185879 < A185877 < A185878 < A185880 < ... See A144112 for the definition of accumulation array.

			Northwest corner:
   1,    5,   16,   40,   85
   3,   17,   56,  140,  295
   6,   38,  128,  320,  670
  10,   70,  240,  600, 1250

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

Cf. A144112, A185877, A185878.
Antidiagonal sums: A037235.
diag (1,5,...): A056108 (4th spoke on hexagonal wheel);
diag (3,11,...): A056106 (2nd spoke on hexagonal wheel);
diag (7,19,...): A003215 (hex numbers);
diag (13,29,...): A144391.

(* This program generates A185878 first and then generates A185880 as the accumulation array of A185878. *)
f[n_,k_]:=(k*n/6)(7-3k+2k^2-3n+3kn);
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185878 *)
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}];
FullSimplify[s[n,k]]
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A185880 *)
f[n_, k_] := (1/72)*k*(1 + k)*n*(1 + n)*(16 - k + 3 *k^2 + 4 *(-1 + k) *n); Table[f[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 21 2017 *)

T(n,k) = C(k,2)*C(n,2)*(3*k^2+4*k*n-k-4*n+16)/18, k>=1, n>=1.

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A185878 Accumulation array of A185877, by antidiagonals.

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A185879 Weight array of A185877, by antidiagonals.

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A185880 Second accumulation array of A185877, by antidiagonals.

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