A244911 Table read by antidiagonals: T(n,k) = n*k + T(n-1,k) for n >=1, T(0,k) = 1.

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 7, 7, 1, 1, 5, 10, 13, 11, 1, 1, 6, 13, 19, 21, 16, 1, 1, 7, 16, 25, 31, 31, 22, 1, 1, 8, 19, 31, 41, 46, 43, 29, 1, 1, 9, 22, 37, 51, 61, 64, 57, 37, 1, 1, 10, 25, 43, 61, 76, 85, 85, 73, 46, 1, 1, 11, 28, 49, 71, 91, 106, 113, 109, 91

Offset

0,5

Comments

T(n,k) is the total number of boxes, when we start with 1 center box (n = 0) then expand 1 box on k-arms for each n iteration. See illustration in links.

It seems that column C(k) = centered k-gonal numbers, and row R(n) = A000217(n)*k + 1.

The triangle under the main diagonal is A121722.

Column N (CN) is the Narayana transform (A001263) of (1, N, 0, 0, 0. Example: C2 (1, 3, 7, 13, ...) is the Narayana transform of (1, 2, 0, 0, 0, ...). - Gary W. Adamson, Oct 01 2015

Links

Table of n, a(n) for n=0..75.

Kival Ngaokrajang, Illustration for n = 0..3, k = 1..4

Formula

T(n,k) = n*k + T(n-1,k) for n >=1, T(0,k) = 1.

Example

Table begins:
       C0  C1  C2  C3  C4  C5
  n/k  0   1   2   3   4   5   ...
R0 0   1   1   1   1   1   1   ...
R1 1   1   2   3   4   5   6   ...
R2 2   1   4   7   10  13  16  ...
R3 3   1   7   13  19  25  31  ...
R4 4   1   11  21  31  41  51  ...
R5 5   1   16  31  46  61  76  ...
R6 6   1   22  43  64  85  106 ...
R7 7   1   29  57  85  113 141 ...
R8 8   1   37  73  109 145 181 ...
R9 9   1   46  91  136 181 226 ...
  ...  ... ... ... ... ... ... ...
C1 = A000124, C2 = A002061, C3 = A005448, C4 = A001844, C5 = A005891, C6 = A003215, C7 = A069099, C8 = A016754, C9 = A060544, C10 = A062786, C11 = A069125, C12  =  A003154.
R1 = A000027, R2 = A016777, R3 = A016921, R4 = A017281, R5 = 15*k + 1, R6 = A215146, R7 = A161714.

Prog

(Small Basic)
For k = 0 to 50
  a[0][k] = 1
  For n = 1 to 50
    a[n][k] = n*k + a[n-1][k]
  EndFor
Endfor
'==================================
For t = 1 to 20
  d = 1
  For nn = 0 To t-1
    kk =  t- d
    TextWindow.Write(a[nn][kk]+", ")
    d = d + 1
  EndFor
Endfor

Crossrefs

Cf. A000217, A121722, A000124, A002061, A005448, A001844, A005891, A003215, A069099, A016754, A060544, A062786, A069125, A003154, A000027, A016777, A016921, A017281, A215146, A161714.

Sequence in context: A113749 A109225 A112564 * A258309 A197957 A089899

Adjacent sequences: A244908 A244909 A244910 * A244912 A244913 A244914

Keyword

nonn,tabl

Author

Kival Ngaokrajang, Jul 07 2014

Status

approved