A066119 -id:A066119 - cp's OEIS frontend

A066120 Superduperprimorials: product of first n terms in Chernoff sequence.

1, 2, 24, 8640, 653184000, 114069441024000000, 598216546007097937920000000000, 1601591599167888308924824752807936000000000000000

Offset: 0

Henry Bottomley, Dec 05 2001

First number with n distinct positive triangular exponents in its prime factorization. Subsequence of A025487. - Matthew Vandermast, Nov 05 2009; edited May 23 2012

			a(4) = 2*(2*(2*3))*(2*(2*3)*(2*3*5)) = 2*(2*6)*(2*6*30) = 2*12*360 = 8640.

Harry J. Smith, Table of n, a(n) for n = 0..17

Cf. A000040, A002110, A006939, A055462, A166475.

a(n) = { my(p=primes(n)); prod(i=1, #p, p[i]^((n - i + 1)*(n - i + 2)/2)) } \\ Harry J. Smith, Feb 01 2010

a(n) = Product_{i=1..n} prime(i)^((n-i+1)*(n-i+2)/2) = a(n-1)*A006939(n) = A066119(n-3, 4).

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

Original entry on oeis.org

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

Henry Bottomley, Dec 05 2001

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

			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)

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

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).

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 *)

(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

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

A066121 Multi-level factorials: triangle with a(n,k)=a(n-1,k-1)*a(n-1,k) but with a(n,1)=n and a(n,n)=1.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 6, 2, 1, 5, 24, 12, 2, 1, 6, 120, 288, 24, 2, 1, 7, 720, 34560, 6912, 48, 2, 1, 8, 5040, 24883200, 238878720, 331776, 96, 2, 1, 9, 40320, 125411328000, 5944066965504000, 79254226206720, 31850496, 192, 2, 1, 10, 362880

Offset: 1

Henry Bottomley, Dec 05 2001

			a(4,2)=a(3,1)*a(3,2)=3*2=6. Rows start 1; 2,1; 3,2,1; 4,6,2,1; ...

Columns include A000027, A000142, A000178, A055462, A057527, A057528. Right hand side includes A000012, A007395, A007283. Cf. A066119.

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A066120 Superduperprimorials: product of first n terms in Chernoff sequence.

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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).

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A066121 Multi-level factorials: triangle with a(n,k)=a(n-1,k-1)*a(n-1,k) but with a(n,1)=n and a(n,n)=1.

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