A267096 -id:A267096 - cp's OEIS frontend

A252738 Row products of irregular table A005940: a(0) = 1; a(1) = 2; for n > 1: 2^(2^(n-2)) * a(n-1) * A003961(a(n-1)); also row products of A163511, A253563, A253565, and A332977.

1, 2, 12, 2160, 2449440000, 8488905214204800000000000, 3025568387202006082882734693673523654400000000000000000000000000

Offset: 0

Antti Karttunen, Dec 21 2014

nonn

			From _Michael De Vlieger_, Jul 21 2023: (Start)
a(0) = 1 = product of {1},
a(1) = 2^1 = product of {2},
a(2) = 2^2 * 3^1 = product of {3, 2^2},
a(3) = 2^4 * 3^3 * 5^1 = product of {5, 2^1*3^1, 3^2, 2^3},
a(4) = 2^8 * 3^7 * 5^4 * 7^1 = product of
  {7, 2^1*5^1, 3^1*5^1, 2^2*3^1, 5^2, 2^1*3^2, 3^3, 2^4},
...
Table of e(n,k) where a(n) = Product_{k=1..n+1} prime(k)^e(n,k):
prime(k)|    2    3    5   7  11  13  17  19 23 29 31 ...
   n\k  |    1    2    3   4   5   6   7   8  9 10 11 ...
   ----------------------------------------------------
    0   |    1
    1   |    2    1
    2   |    4    3    1
    3   |    8    7    4   1
    4   |   16   15   11   5   1
    5   |   32   31   26  16   6   1
    6   |   64   63   57  42  22   7   1
    7   |  128  127  120  99  64  29   8   1
    8   |  256  255  247 219 163  93  37   9  1
    9   |  512  511  502 466 382 256 130  46 10  1
   10   | 1024 1023 1013 968 848 638 386 176 56 11  1
  ... (End)

Alois P. Heinz, Table of n, a(n) for n = 0..9

These are row products of irregular tables A005940, A163511, A253563 and A253565, which all are shaped like a binary tree.
Partial products of A252740.
Cf. A252737 (row sums), A252739 (divided by n), A252741 (divided by n!).
Cf. A000079, A000225, A000290, A003961, A163511, A191555, A266639, A267096, A332977.

Table[Times @@ Array[Prime[# + 1]^Sum[Binomial[n, # + j], {j, 0, n}] &, n + 1, 0], {n, 0, 5}] (* Michael De Vlieger, Jul 21 2023 *)

allocatemem(234567890);
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A252738print(up_to_n) = { my(s, i=0, n=0); for(n=0, up_to_n, if(0 == n, s = 1, if(1 == n, s = 2; lev = vector(1); lev[1] = 2, oldlev = lev; lev = vector(2*length(oldlev)); s = 1; for(i = 0, (2^(n-1))-1, lev[i+1] = if((i%2),A003961(oldlev[(i\2)+1]),2*oldlev[(i\2)+1]); s *= lev[i+1]))); write("b252738.txt", n, " ", s)); }; \\ Counts them empirically.
A252738print(7);

(definec (A252738rec n) (if (<= n 1) (+ 1 n) (* (A000079 (A000079 (- n 2))) (A252738rec (- n 1)) (A003961 (A252738rec (- n 1)))))) ;; Implements the given recurrence; uses the memoizing definec-macro.
(define (A252738 n) (if (zero? n) 1 (mul A163511 (A000079 (- n 1)) (A000225 n))))
(define (mul intfun lowlim uplim) (let multloop ((i lowlim) (res 1)) (cond ((> i uplim) res) (else (multloop (+ 1 i) (* res (intfun i)))))))
;; Another alternative, implementing the new recurrence:
(definec (A252738 n) (if (<= n 1) (+ 1 n) (* (A267096 (- n 2)) (A000290 (A252738 (- n 1)))))) ;; Antti Karttunen, Feb 06 2016

a(0) = 1; a(1) = 2; for n > 1: a(n) = 2^(2^(n-2)) * a(n-1) * A003961(a(n-1)).
a(0) = 1; for n>=1: a(n) = Product_{k=A000079(n-1) .. A000225(n)} A163511(k) = Product_{k=2^(n-1) .. (2^n)-1} A163511(k).
a(0) = 1; a(1) = 2; for n > 1: a(n) = A267096(n-2) * a(n-1)^2. [Compare to the formulas of A191555] - Antti Karttunen, Feb 06 2016
From Michael De Vlieger, Jul 21 2023: (Start)
a(n) = Product_{k=1..n+1} prime(k)^e(n,k), where e(n,k) = k-th term in row n of A055248.
A067255(a(n)) = row n of A055248. (End)

Typos in the second formula corrected by Antti Karttunen, Feb 06 2016

A007188 Multiplicative encoding of Pascal triangle: Product p(i+1)^C(n,i).

Original entry on oeis.org

2, 6, 90, 47250, 66852843750, 2806877704512541816406250, 1216935896582703898519354781702537118597533386230468750

Offset: 0

N. J. A. Sloane

nonn

n-th power of x+1 using the encoding of polynomials defined in A206284 and A297845. - Peter Munn, Jul 20 2022

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
N. J. A. Sloane, An on-line version of the Encyclopedia of Integer Sequences, Electronic J. Combinatorics, Vol. 1, no. 1, 1994.
Index entries for triangles and arrays related to Pascal's triangle

Leftmost column of square array A066117.

Cf. A007318, A123098, A206295, A206284, A267096, A297845.

c[n_] := CoefficientList[(1 + x)^n, x]; f[n_] := Product[Prime[k]^c[n][[k]], {k, 1, Length[c[n]]}]; Table[f[n], {n, 1, 7}] (* Clark Kimberling, Feb 05 2012 *)

a(0) = 2; for n > 0, a(n) = A297845(a(n-1), 6). - Peter Munn, Jul 20 2022

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

A276804 Second column T[.,2] of array T = A255483: T[0,j] = prime(j), T[i+1,j] = T[i,j]*T[i,j+1]/gcd(T[i,j],T[i,j+1])^2, i >= 0, j >= 1.

Original entry on oeis.org

3, 15, 21, 1155, 39, 3315, 5187, 111546435, 87, 13485, 22533, 1575169365, 48633, 6022953885, 12684118629, 961380175077106319535, 183, 61305, 90951, 24466273755, 187941, 88836891585, 157950690807, 133754519645521334494935, 536007, 573342567585

Offset: 0

M. F. Hasler, Sep 17 2016

nonn

By construction all terms are divisible by 3, and the n-th term a(n-1) is divisible by prime(n+1). We have a(n)/3 = (1, 5, 7, 385, 13, 1105, 1729, 37182145, 29, 4495, ...). Neither the sequence of primes appearing here, (5, 7, 13, 29, 61, ...), nor its complement in the primes, ([2, 3,] 11, 17, 19, 23, 31, 37, 41, 43, 47, 53, 59, 67, ...), seem to be listed in the OEIS.

This is also the multiplicative encoding of Pascal's triangle in Z_2 (A047999), shifted by prefixing an initial 0 to the n-th row; e.g., n=2 => 1,0,1 => 0,1,0,1 => 2^0 * 3^1 * 5^0 * 7^1 = a(2).

Cf. A255483 (the square array T), A123098 (first column of T), A003961.

Cf. A007913, A252738, A267096.

A276804(n)=prod(j=0, n, if(bitand(n-j, j), 1, prime(j+2)))

a(n) = A003961(A123098(n)).
a(n) = Prod_{j=0..n} prime(j+2)^(!(n-j & j)), where ! is "not" (=0 for nonzero and 1 for zero) and & is bitwise AND.
a(n) = A007913(A267096(n)) = A007913(A252738(n+2)). - Antti Karttunen, Sep 18 2016

cp's OEIS Frontend

A252738 Row products of irregular table A005940: a(0) = 1; a(1) = 2; for n > 1: 2^(2^(n-2)) * a(n-1) * A003961(a(n-1)); also row products of A163511, A253563, A253565, and A332977.

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A007188 Multiplicative encoding of Pascal triangle: Product p(i+1)^C(n,i).

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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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A276804 Second column T[.,2] of array T = A255483: T[0,j] = prime(j), T[i+1,j] = T[i,j]*T[i,j+1]/gcd(T[i,j],T[i,j+1])^2, i >= 0, j >= 1.

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A266639 a(n) = A252738(n) / A191555(n).

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