A185912 -id:A185912 - cp's OEIS frontend

A185913 Accumulation array of A185912, by antidiagonals.

1, 4, 6, 10, 21, 20, 20, 48, 66, 50, 35, 90, 144, 160, 105, 56, 150, 260, 340, 330, 196, 84, 231, 420, 600, 690, 609, 336, 120, 336, 630, 950, 1200, 1260, 1036, 540, 165, 468, 896, 1400, 1875, 2170, 2128, 1656, 825, 220, 630, 1224, 1960, 2730, 3360, 3640, 3384, 2520, 1210, 286, 825, 1620, 2640, 3780, 4851, 5600, 5760, 5130, 3685, 1716, 364, 1056, 2090, 3450, 5040, 6664, 8036, 8820, 8700, 7480, 5214, 2366, 455

Offset: 1

Clark Kimberling, Feb 06 2011

A member of the accumulation chain ... < A185910 < A185911 < A185912 < A185913 < ...

(See A144112 for definitions of weight array and accumulation array.)

			Northwest corner:
1.....4.....10.....20.....35
6.....21....48.....90.....150
20....66....144....260....420
50....160...340....600....950

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

Cf. A144112, A185912.
Row 1 to 2: A000292, 3*A005581.
Column 1: A002415.

(* The program generates A185912 and its accumulation array A185913 *)
f[n_,k_]:=(k*n/6)(-2+3k+3n+2n^2);
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]  (* array A185912 *)
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]]  (* formula for A185913 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* array A185913 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

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

A185910 Array: T(n,k) = n^2 + k - 1, by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 5, 9, 4, 6, 10, 16, 5, 7, 11, 17, 25, 6, 8, 12, 18, 26, 36, 7, 9, 13, 19, 27, 37, 49, 8, 10, 14, 20, 28, 38, 50, 64, 9, 11, 15, 21, 29, 39, 51, 65, 81, 10, 12, 16, 22, 30, 40, 52, 66, 82, 100, 11, 13, 17, 23, 31, 41, 53, 67, 83, 101, 121, 12, 14, 18, 24, 32, 42, 54, 68, 84, 102, 122, 144, 13, 15, 19, 25, 33, 43, 55, 69, 85, 103, 123, 145, 169, 14, 16, 20, 26, 34, 44, 56, 70, 86, 104, 124, 146, 170, 196

Offset: 1

Clark Kimberling, Feb 06 2011

A member of the accumulation chain ... < A185911 < A185910 < A185912 < A185913 < ... (See A144112 for definitions of weight array and accumulation array.)

			Northwest corner:
   1,  2,  3,  4,  5
   4,  5,  6,  7,  8
   9, 10, 11, 12, 13
  16, 17, 18, 19, 20

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

Cf. A144112, A185910, A185912.

(* This program generates the array A185910, its accumulation array A185812, and its weight array A185911. *)
f[n_,0]:=0;f[0,k_]:=0;
f[n_,k_]:=n^2+k-1;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185910 *)
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 A185812 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]
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}]] (* A185911 *)
Table[w[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

T(n,k) = n^2 + k - 1, k >= 1, n >= 1.

A185911 Weight array of A185910, by antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 0, 5, 1, 0, 0, 7, 1, 0, 0, 0, 9, 1, 0, 0, 0, 0, 11, 1, 0, 0, 0, 0, 0, 13, 1, 0, 0, 0, 0, 0, 0, 15, 1, 0, 0, 0, 0, 0, 0, 0, 17, 1, 0, 0, 0, 0, 0, 0, 0, 0, 19, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 23, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 27

Offset: 1

Clark Kimberling, Feb 06 2011

A member of the accumulation chain ... < A185910 < A185911 < A185912 < A185913 < ...

(See A144112 for definitions of weight array and accumulation array.)

			Northwest corner:
  1, 1, 1, 1, 1, 1
  3, 0, 0, 0, 0, 0
  5, 0, 0, 0, 0, 0
  7, 0, 0, 0, 0, 0
  9, 0, 0, 0, 0, 0

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

Cf. A144112, A185910, A185912.

f[n_, 0] := 0; f[0, k_] := 0; f[n_, k_] := n^2 + k - 1;
w[m_, n_] := f[m, n] + f[m - 1, n - 1] - f[m, n - 1] - f[m - 1, n] /; Or[m > 0, n > 0];
Table[w[n - k + 1, k], {n, 50}, {k, n, 1, -1}] // Flatten
T[1, k_] := 1; T[n_, 1] := 2*n - 1; T[n_, k_] := 0; Table[T[n - k + 1, k], {n, 10}, {k, n, 1, -1}]//Flatten (* G. C. Greubel, Jul 22 2017 *)

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

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A185913 Accumulation array of A185912, by antidiagonals.

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A185910 Array: T(n,k) = n^2 + k - 1, by antidiagonals.

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A185911 Weight array of A185910, by antidiagonals.

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