A066117 - OEIS

%I #24 Jul 27 2022 16:36:26

%S 2,3,6,5,15,90,7,35,525,47250,11,77,2695,1414875,66852843750,13,143,

%T 11011,29674645,41985913344375,2806877704512541816406250,17,221,31603,

%U 347980633,10326201751150285,433555011900329243987584396875

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

%C 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

%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>

%F From _Antti Karttunen_, Sep 19 2016: (Start)

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

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

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

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

%F (End)

%F 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

%e T(4,3) = T(3,2)*T(4,2) = 15*35 = 525. Rows start

%e      2;

%e     3, 6;

%e   5, 15, 90;

%e 7, 35, 525, 47250;

%e ...

%e From _Antti Karttunen_, Sep 18 2016: (Start)

%e Alternatively, this table can be viewed as a square array. Then the top left 5x4 corner looks as:

%e     2,       3,        5,         7,         11

%e     6,      15,       35,        77,        143

%e    90,     525,     2695,     11011,      31603

%e 47250, 1414875, 29674645, 347980633, 2255916949

%e (End)

%t T[n_, 1] := Prime[n];

%t T[n_, k_] := T[n, k] = T[n - 1, k - 1]*T[n, k - 1];

%t Table[T[n, k], {n, 1, 7}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Nov 13 2017 *)

%o (Scheme)

%o (define (A066117 n) (A066117bi (A002260 n) (A004736 n)))

%o ;; Compute as a square array, with row >= 1, col >= 1:

%o (define (A066117bi row col) (if (= 1 row) (A000040 col) (* (A066117bi (- row 1) col) (A066117bi (- row 1) (+ col 1)))))

%o ;; With alternative recurrence:

%o (define (A066117bi row col) (if (= 1 col) (A007188 (- row 1)) (A003961 (A066117bi row (- col 1)))))

%o ;; _Antti Karttunen_, Sep 18 2016

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

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

%Y Cf. A003961, A055396, A297845.

%Y Cf. A064319, A066119.

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

%K nonn,tabl

%O 1,1

%A _Henry Bottomley_, Dec 05 2001