A323956 Triangle read by rows: T(n, k) = 1 + n * (n - k) for 1 <= k <= n.

1, 3, 1, 7, 4, 1, 13, 9, 5, 1, 21, 16, 11, 6, 1, 31, 25, 19, 13, 7, 1, 43, 36, 29, 22, 15, 8, 1, 57, 49, 41, 33, 25, 17, 9, 1, 73, 64, 55, 46, 37, 28, 19, 10, 1, 91, 81, 71, 61, 51, 41, 31, 21, 11, 1, 111, 100, 89, 78, 67, 56, 45, 34, 23, 12, 1

Offset: 1

Author

Gus Wiseman, Feb 10 2019

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  n\k:   1   2   3   4   5   6   7   8   9  10  11  12
  ====================================================
    1:   1
    2:   3   1
    3:   7   4   1
    4:  13   9   5   1
    5:  21  16  11   6   1
    6:  31  25  19  13   7   1
    7:  43  36  29  22  15   8   1
    8:  57  49  41  33  25  17   9   1
    9:  73  64  55  46  37  28  19  10   1
   10:  91  81  71  61  51  41  31  21  11   1
   11: 111 100  89  78  67  56  45  34  23  12   1
   12: 133 121 109  97  85  73  61  49  37  25  13   1
  etc.
		

Links

• G. C. Greubel, Rows n = 1..100 of triangle, flattened

Crossrefs

First column is A002061. Second column is A000290. Third column is A028387.

Cf. A000126, A001610, A001644, A006000, A169985, A306351, A323952, A323955.

Programs

Magma

[[1+n*(n-k): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Apr 22 2019

Mathematica

Table[1+n*(n-k),{n,12},{k,n}]//Flatten

PARI

{T(n,k) = 1+n*(n-k)}; \\ G. C. Greubel, Apr 22 2019

Sage

[[1+n*(n-k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Apr 22 2019

Formula

From Werner Schulte, Feb 12 2019: (Start)

G.f.: Sum_{n>0,k=1..n} T(n,k)*x^k*t^n = x*t*((1-t+2*t^2)*(1-x*t) + (1-t)*t)/((1-t)^3*(1-x*t)^2).

Row sums: Sum_{k=1..n} T(n,k) = A006000(n-1) for n > 0.

Recurrence: T(n,k) = T(n,k-1) - n for 1 < k <= n with initial values T(n,1) = n^2-n+1 for n > 0.

Recurrence: T(n,k) = T(n-1,k) + 2*n-k-1 for 1 <= k < n with initial values T(n,n) = 1 for n > 0.

(End)