A066117 Triangle read by rows: T(n,k) = T(n-1,k-1)*T(n,k-1) and T(n,1) = prime(n).

2, 3, 6, 5, 15, 90, 7, 35, 525, 47250, 11, 77, 2695, 1414875, 66852843750, 13, 143, 11011, 29674645, 41985913344375, 2806877704512541816406250, 17, 221, 31603, 347980633, 10326201751150285, 433555011900329243987584396875

Offset: 1

Author

Henry Bottomley, Dec 05 2001

Keywords

• nonn
• tabl

Comments

As a square array read by descending antidiagonals, A(n, k), n >= 1, k >= 1, gives the encoding defined in A297845 of the polynomial (x+1)^(n-1) * x^(k-1). - Peter Munn, Jul 27 2022

Examples

			T(4,3) = T(3,2)*T(4,2) = 15*35 = 525. Rows start
     2;
    3, 6;
  5, 15, 90;
7, 35, 525, 47250;
...
From _Antti Karttunen_, Sep 18 2016: (Start)
Alternatively, this table can be viewed as a square array. Then the top left 5x4 corner looks as:
    2,       3,        5,         7,         11
    6,      15,       35,        77,        143
   90,     525,     2695,     11011,      31603
47250, 1414875, 29674645, 347980633, 2255916949
(End)
		

Links

• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A000040, A006094 and A066116 (three leftmost diagonal of triangular table = three topmost rows of square array).

Cf. A007188, A267096 (two rightmost diagonals of the triangular table = two leftmost columns of square array).

Cf. A003961, A055396, A297845.

Cf. A064319, A066119.

Cf. also A099884, A255483, A276586, A276588 (other arrays derived from this one).

Programs

Mathematica

T[n_, 1] := Prime[n];
T[n_, k_] := T[n, k] = T[n - 1, k - 1]*T[n, k - 1];
Table[T[n, k], {n, 1, 7}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 13 2017 *)

Scheme

(define (A066117 n) (A066117bi (A002260 n) (A004736 n)))
;; Compute as a square array, with row >= 1, col >= 1:
(define (A066117bi row col) (if (= 1 row) (A000040 col) (* (A066117bi (- row 1) col) (A066117bi (- row 1) (+ col 1)))))
;; With alternative recurrence:
(define (A066117bi row col) (if (= 1 col) (A007188 (- row 1)) (A003961 (A066117bi row (- col 1)))))
;; Antti Karttunen, Sep 18 2016

Formula

From Antti Karttunen, Sep 19 2016: (Start)

When computed as a square array A(row,col), row >= 1, col >= 1:

A(1,col) = A000040(col), for row > 1, A(row,col) = A(row-1,col)*A(row-1,col+1).

A(row,1) = A007188(row-1), for col > 1, A(row,col) = A003961(A(row,col-1)).

For all row >= 1, col >= 1, A055396(A(row,col)) = col.

(End)

A(1,1) = 2; for n > 1, A(n,k) = A297845(A(n-1,k),6); for k > 1, A(n,k) = A297845(A(n,k-1),3). - Peter Munn, Jul 20 2022