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

%I A066117 #24 Jul 27 2022 16:36:26
%S A066117 2,3,6,5,15,90,7,35,525,47250,11,77,2695,1414875,66852843750,13,143,
%T A066117 11011,29674645,41985913344375,2806877704512541816406250,17,221,31603,
%U A066117 347980633,10326201751150285,433555011900329243987584396875
%N A066117 Triangle read by rows: T(n,k) = T(n-1,k-1)*T(n,k-1) and T(n,1) = prime(n).
%C A066117 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 A066117 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A066117 From _Antti Karttunen_, Sep 19 2016: (Start)
%F A066117 When computed as a square array A(row,col), row >= 1, col >= 1:
%F A066117 A(1,col) = A000040(col), for row > 1, A(row,col) = A(row-1,col)*A(row-1,col+1).
%F A066117 A(row,1) = A007188(row-1), for col > 1, A(row,col) = A003961(A(row,col-1)).
%F A066117 For all row >= 1, col >= 1, A055396(A(row,col)) = col.
%F A066117 (End)
%F A066117 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 A066117 T(4,3) = T(3,2)*T(4,2) = 15*35 = 525. Rows start
%e A066117      2;
%e A066117     3, 6;
%e A066117   5, 15, 90;
%e A066117 7, 35, 525, 47250;
%e A066117 ...
%e A066117 From _Antti Karttunen_, Sep 18 2016: (Start)
%e A066117 Alternatively, this table can be viewed as a square array. Then the top left 5x4 corner looks as:
%e A066117     2,       3,        5,         7,         11
%e A066117     6,      15,       35,        77,        143
%e A066117    90,     525,     2695,     11011,      31603
%e A066117 47250, 1414875, 29674645, 347980633, 2255916949
%e A066117 (End)
%t A066117 T[n_, 1] := Prime[n];
%t A066117 T[n_, k_] := T[n, k] = T[n - 1, k - 1]*T[n, k - 1];
%t A066117 Table[T[n, k], {n, 1, 7}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Nov 13 2017 *)
%o A066117 (Scheme)
%o A066117 (define (A066117 n) (A066117bi (A002260 n) (A004736 n)))
%o A066117 ;; Compute as a square array, with row >= 1, col >= 1:
%o A066117 (define (A066117bi row col) (if (= 1 row) (A000040 col) (* (A066117bi (- row 1) col) (A066117bi (- row 1) (+ col 1)))))
%o A066117 ;; With alternative recurrence:
%o A066117 (define (A066117bi row col) (if (= 1 col) (A007188 (- row 1)) (A003961 (A066117bi row (- col 1)))))
%o A066117 ;; _Antti Karttunen_, Sep 18 2016
%Y A066117 Cf. A000040, A006094 and A066116 (three leftmost diagonal of triangular table = three topmost rows of square array).
%Y A066117 Cf. A007188, A267096 (two rightmost diagonals of the triangular table = two leftmost columns of square array).
%Y A066117 Cf. A003961, A055396, A297845.
%Y A066117 Cf. A064319, A066119.
%Y A066117 Cf. also A099884, A255483, A276586, A276588 (other arrays derived from this one).
%K A066117 nonn,tabl
%O A066117 1,1
%A A066117 _Henry Bottomley_, Dec 05 2001