A123098 -id:A123098 - cp's OEIS frontend

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1

Offset: 0

N. J. A. Sloane

Restored the alternative spelling of Sierpinski to facilitate searching for this triangle using regular-expression matching commands in ASCII. - N. J. A. Sloane, Jan 18 2016
Also triangle giving successive states of cellular automaton generated by "Rule 60" and "Rule 102". - Hans Havermann, May 26 2002
Also triangle formed by reading triangle of Eulerian numbers (A008292) mod 2. - Philippe Deléham, Oct 02 2003
Self-inverse when regarded as an infinite lower triangular matrix over GF(2).
Start with [1], repeatedly apply the map 0 -> [00/00], 1 -> [10/11] [Allouche and Berthe]
Also triangle formed by reading triangles A011117, A028338, A039757, A059438, A085881, A086646, A086872, A087903, A104219 mod 2. - Philippe Deléham, Jun 18 2005
J. H. Conway writes (in Math Forum): at least the first 31 rows give odd-sided constructible polygons (sides 1, 3, 5, 15, 17, ... see A001317). The 1's form a Sierpiński sieve. - M. Dauchez (mdzzdm(AT)yahoo.fr), Sep 19 2005
When regarded as an infinite lower triangular matrix, its inverse is a (-1,0,1)-matrix with zeros undisturbed and the nonzero entries in every column form the Prouhet-Thue-Morse sequence (1,-1,-1,1,-1,1,1,-1,...) A010060 (up to relabeling). - David Callan, Oct 27 2006
Triangle read by rows: antidiagonals of an array formed by successive iterates of running sums mod 2, beginning with (1, 1, 1, ...). - Gary W. Adamson, Jul 10 2008
T(n,k) = A057427(A143333(n,k)). - Reinhard Zumkeller, Oct 24 2010
The triangle sums, see A180662 for their definitions, link Sierpiński’s triangle A047999 with seven sequences, see the crossrefs. The Kn1y(n) and Kn2y(n), y >= 1, triangle sums lead to the Sierpiński-Stern triangle A191372. - Johannes W. Meijer, Jun 05 2011
Used to compute the total Steifel-Whitney cohomology class of the Real Projective space. This was an essential component of the proof that there are no product operations without zero divisors on R^n for n not equal to 1, 2, 4 or 8 (real numbers, complex numbers, quaternions, Cayley numbers), proved by Bott and Milnor. - Marcus Jaiclin, Feb 07 2012
T(n,k) = A134636(n,k) mod 2. - Reinhard Zumkeller, Nov 23 2012
T(n,k) = 1 - A219463(n,k), 0 <= k <= n. - Reinhard Zumkeller, Nov 30 2012
From Vladimir Shevelev, Dec 31 2013: (Start)
Also table of coefficients of polynomials s_n(x) of degree n which are defined by formula s_n(x) = Sum_{i=0..n} (binomial(n,i) mod 2)*x^k. These polynomials we naturally call Sierpiński's polynomials. They also are defined by the recursion: s_0(x)=1, s_(2*n+1)(x) = (x+1)*s_n(x^2), n>=0, and s_(2*n)(x) = s_n(x^2), n>=1.
Note that: s_n(1)  = A001316(n),
           s_n(2)  = A001317(n),
           s_n(3)  = A100307(n),
           s_n(4)  = A001317(2*n),
           s_n(5)  = A100308(n),
           s_n(6)  = A100309(n),
           s_n(7)  = A100310(n),
           s_n(8)  = A100311(n),
           s_n(9)  = A100307(2*n),
           s_n(10) = A006943(n),
           s_n(16) = A001317(4*n),
           s_n(25) = A100308(2*n), etc.
The equality s_n(10) = A006943(n) means that sequence A047999 is obtained from A006943 by the separation by commas of the digits of its terms. (End)
Comment from N. J. A. Sloane, Jan 18 2016: (Start)
Take a diamond-shaped region with edge length n from the top of the triangle, and rotate it by 45 degrees to get a square S_n. Here is S_6:
  [1, 1, 1, 1, 1, 1]
  [1, 0, 1, 0, 1, 0]
  [1, 1, 0, 0, 1, 1]
  [1, 0, 0, 0, 1, 0]
  [1, 1, 1, 1, 0, 0]
  [1, 0, 1, 0, 0, 0].
Then (i) S_n contains no square (parallel to the axes) with all four corners equal to 1 (cf. A227133); (ii) S_n can be constructed by using the greedy algorithm with the constraint that there is no square with that property; and (iii) S_n contains A064194(n) 1's. Thus A064194(n) is a lower bound on A227133(n). (End)
See A123098 for a multiplicative encoding of the rows, i.e., product of the primes selected by nonzero terms; e.g., 1 0 1 => 2^1 * 3^0 * 5^1. - M. F. Hasler, Sep 18 2016
From Valentin Bakoev, Jul 11 2020: (Start)
The Sierpinski's triangle with 2^n rows is a part of a lower triangular matrix M_n of dimension 2^n X 2^n. M_n is a block matrix defined recursively: M_1= [1, 0], [1, 1], and for n>1, M_n = [M_(n-1), O_(n-1)], [M_(n-1), M_(n-1)], where M_(n-1) is a block matrix of the same type, but of dimension 2^(n-1) X 2^(n-1), and O_(n-1) is the zero matrix of dimension 2^(n-1) X 2^(n-1). Here is how M_1, M_2 and M_3 look like:
1 0     1 0 0 0      1 0 0 0 0 0 0 0
1 1     1 1 0 0      1 1 0 0 0 0 0 0   - It is seen the self-similarity of the
        1 0 1 0      1 0 1 0 0 0 0 0     matrices M_1, M_2, ..., M_n, ...,
        1 1 1 1      1 1 1 1 0 0 0 0     analogously to the Sierpinski's fractal.
                     1 0 0 0 1 0 0 0
                     1 1 0 0 1 1 0 0
                     1 0 1 0 1 0 1 0
                     1 1 1 1 1 1 1 1
M_n can also be defined as M_n = M_1 X M_(n-1) where X denotes the Kronecker product. M_n is an important matrix in coding theory, cryptography, Boolean algebra, monotone Boolean functions, etc. It is a transformation matrix used in computing the algebraic normal form of Boolean functions. Some properties and links concerning M_n can be seen in LINKS. (End)
Sierpinski's gasket has fractal (Hausdorff) dimension log(A000217(2))/log(2) = log(3)/log(2) = 1.58496... (and cf. A020857). This gasket is the first of a family of gaskets formed by taking the Pascal triangle (A007318) mod j, j >= 2 (see CROSSREFS). For prime j, the dimension of the gasket is log(A000217(j))/log(j) = log(j(j + 1)/2)/log(j) (see Reiter and Bondarenko references). - Richard L. Ollerton, Dec 14 2021

			Triangle begins:
              1,
             1,1,
            1,0,1,
           1,1,1,1,
          1,0,0,0,1,
         1,1,0,0,1,1,
        1,0,1,0,1,0,1,
       1,1,1,1,1,1,1,1,
      1,0,0,0,0,0,0,0,1,
     1,1,0,0,0,0,0,0,1,1,
    1,0,1,0,0,0,0,0,1,0,1,
   1,1,1,1,0,0,0,0,1,1,1,1,
  1,0,0,0,1,0,0,0,1,0,0,0,1,
  ...

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
Brand, Neal; Das, Sajal; Jacob, Tom. The number of nonzero entries in recursively defined tables modulo primes. Proceedings of the Twenty-first Southeastern Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1990). Congr. Numer. 78 (1990), 47--59. MR1140469 (92h:05004).
John W. Milnor and James D. Stasheff, Characteristic Classes, Princeton University Press, 1974, pp. 43-49 (sequence appears on p. 46).
H.-O. Peitgen, H. Juergens and D. Saupe: Chaos and Fractals (Springer-Verlag 1992), p. 408.
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.

N. J. A. Sloane, Table of n, a(n) for n = 0..10584 [First 144 rows, flattened; first 50 rows from T. D. Noe].
J.-P. Allouche and V. Berthe, Triangle de Pascal, complexité et automates, Bulletin of the Belgian Mathematical Society Simon Stevin 4.1 (1997): 1-24.
J.-P. Allouche, F. v. Haeseler, H.-O. Peitgen and G. Skordev, Linear cellular automata, finite automata and Pascal's triangle, Discrete Appl. Math. 66 (1996), 1-22.
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.],
J. Baer, Explore patterns in Pascal's Triangle
Valentin Bakoev, Fast Bitwise Implementation of the Algebraic Normal Form Transform, Serdica J. of Computing 11 (2017), No 1, 45-57.
Valentin Bakoev, Properties and links concerning M_n
Thomas Baruchel, Flattening Karatsuba's Recursion Tree into a Single Summation, SN Computer Science (2020) Vol. 1, Article No. 48.
Thomas Baruchel, A non-symmetric divide-and-conquer recursive formula for the convolution of polynomials and power series, arXiv:1912.00452 [math.NT], 2019.
A. Bogomolny, Dot Patterns and Sierpinski Gasket
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see pp. 130-132.
Paul Bradley and Peter Rowley, Orbits on k-subsets of 2-transitive Simple Lie-type Groups, 2014.
E. Burlachenko, Fractal generalized Pascal matrices, arXiv:1612.00970 [math.NT], 2016. See p. 9.
S. Butkevich, Pascal Triangle Applet
David Callan, Sierpinski's triangle and the Prouhet-Thue-Morse word, arXiv:math/0610932 [math.CO], 2006.
B. Cherowitzo, Pascal's Triangle using Clock Arithmetic, Part I
B. Cherowitzo, Pascal's Triangle using Clock Arithmetic, Part II
C. Cobeli, A. Zaharescu, A game with divisors and absolute differences of exponents, arXiv:1411.1334 [math.NT], 2014; Journal of Difference Equations and Applications, Vol. 20, #11, 2014.
Ilya Gutkovskiy, Illustrations (triangle formed by reading Pascal's triangle mod m)
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
Brady Haran, Chaos Game, Numberphile video, YouTube (April 27, 2017).
I. Kobayashi et al., Pascal's Triangle
Dr. Math, Regular polygon formulas [Broken link?]
Y. Moshe, The distribution of elements in automatic double sequences, Discr. Math., 297 (2005), 91-103.
National Curve Bank, Sierpinski Triangles
Hieu D. Nguyen, A Digital Binomial Theorem, arXiv:1412.3181 [math.NT], 2014.
S. Northshield, Sums across Pascal's triangle modulo 2, Congressus Numerantium, 200, pp. 35-52, 2010.
A. M. Reiter, Determining the dimension of fractals generated by Pascal's triangle, Fibonacci Quarterly, 31(2), 1993, pp. 112-120.
F. Richman, Javascript for computing Pascal's triangle modulo n. Go to this page, then under "Modern Algebra and Other Things", click "Pascal's triangle modulo n".
Vladimir Shevelev, On Stephan's conjectures concerning Pascal triangle modulo 2 and their polynomial generalization, J. of Algebra Number Theory: Advances and Appl., 7 (2012), no.1, 11-29. Also arXiv:1011.6083, 2010.
N. J. A. Sloane, Illustration of rows 0 to 32 (encoignure style)
N. J. A. Sloane, Illustration of rows 0 to 64 (encoignure style)
N. J. A. Sloane, Illustration of rows 0 to 128 (encoignure style)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Eric Weisstein's World of Mathematics, Sierpiński Sieve, Rule 60, Rule 102
Index entries for sequences related to cellular automata
Index entries for triangles and arrays related to Pascal's triangle
Index entries for sequences generated by sieves

Sequences based on the triangles formed by reading Pascal's triangle mod m: (this sequence) (m = 2), A083093 (m = 3), A034931 (m = 4), A095140 (m = 5), A095141 (m = 6), A095142 (m = 7), A034930(m = 8), A095143 (m = 9), A008975 (m = 10), A095144 (m = 11), A095145 (m = 12), A275198 (m = 14), A034932 (m = 16).
Other versions: A090971, A038183.
Cf. A007318, A054431, A001317, A008292, A083093, A034931, A034930, A008975, A034932, A166360, A249133, A064194, A227133.
From Johannes W. Meijer, Jun 05 2011: (Start)
A106344 is a skew version of this triangle.
Triangle sums (see the comments): A001316 (Row1; Related to Row2), A002487 (Related to Kn11, Kn12, Kn13, Kn21, Kn22, Kn23), A007306 (Kn3, Kn4), A060632 (Fi1, Fi2), A120562 (Ca1, Ca2), A112970 (Gi1, Gi2), A127830 (Ze3, Ze4). (End)

import Data.Bits (xor)
a047999 :: Int -> Int -> Int
a047999 n k = a047999_tabl !! n !! k
a047999_row n = a047999_tabl !! n
a047999_tabl = iterate (\row -> zipWith xor ([0] ++ row) (row ++ [0])) [1]
-- Reinhard Zumkeller, Dec 11 2011, Oct 24 2010

A047999:= func< n,k | BitwiseAnd(n-k, k) eq 0 select 1 else 0 >;
[A047999(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 03 2024

# Maple code for first M rows (here M=10) - N. J. A. Sloane, Feb 03 2016
ST:=[1,1,1]; a:=1; b:=2; M:=10;
for n from 2 to M do ST:=[op(ST),1];
for i from a to b-1 do ST:=[op(ST), (ST[i+1]+ST[i+2]) mod 2 ]; od:
ST:=[op(ST),1];
a:=a+n; b:=a+n; od:
ST; # N. J. A. Sloane
# alternative
A047999 := proc(n,k)
    modp(binomial(n,k),2) ;
end proc:
seq(seq(A047999(n,k),k=0..n),n=0..12) ; # R. J. Mathar, May 06 2016

Mod[ Flatten[ NestList[ Prepend[ #, 0] + Append[ #, 0] &, {1}, 13]], 2] (* Robert G. Wilson v, May 26 2004 *)
rows = 14; ca = CellularAutomaton[60, {{1}, 0}, rows-1]; Flatten[ Table[ca[[k, 1 ;; k]], {k, 1, rows}]] (* Jean-François Alcover, May 24 2012 *)
Mod[#,2]&/@Flatten[Table[Binomial[n,k],{n,0,20},{k,0,n}]] (* Harvey P. Dale, Jun 26 2019 *)
A047999[n_,k_]:= Boole[BitAnd[n-k,k]==0];
Table[A047999[n,k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 03 2025 *)

\\ Recurrence for Pascal's triangle mod p, here p = 2.
p = 2; s=13; T=matrix(s,s); T[1,1]=1;
for(n=2,s, T[n,1]=1; for(k=2,n, T[n,k] = (T[n-1,k-1] + T[n-1,k])%p ));
for(n=1,s,for(k=1,n,print1(T[n,k],", "))) \\ Gerald McGarvey, Oct 10 2009

A011371(n)=my(s);while(n>>=1,s+=n);s
T(n,k)=A011371(n)==A011371(k)+A011371(n-k) \\ Charles R Greathouse IV, Aug 09 2013

T(n,k)=bitand(n-k,k)==0 \\ Charles R Greathouse IV, Aug 11 2016

def A047999_T(n,k):
    return int(not ~n & k) # Chai Wah Wu, Feb 09 2016

Lucas's Theorem is that T(n,k) = 1 if and only if the 1's in the binary expansion of k are a subset of the 1's in the binary expansion of n; or equivalently, k AND NOT n is zero, where AND and NOT are bitwise operators. - Chai Wah Wu, Feb 09 2016 and N. J. A. Sloane, Feb 10 2016
Sum_{k>=0} T(n, k) = A001316(n) = 2^A000120(n).
T(n,k) = T(n-1,k-1) XOR T(n-1,k), 0 < k < n; T(n,0) = T(n,n) = 1. - Reinhard Zumkeller, Dec 13 2009
T(n,k) = (T(n-1,k-1) + T(n-1,k)) mod 2 = |T(n-1,k-1) - T(n-1,k)|, 0 < k < n; T(n,0) = T(n,n) = 1. - Rick L. Shepherd, Feb 23 2018
From Vladimir Shevelev, Dec 31 2013: (Start)
For polynomial {s_n(x)} we have
s_0(x)=1; for n>=1, s_n(x) = Product_{i=1..A000120(n)} (x^(2^k_i) + 1),
if the binary expansion of n is n = Sum_{i=1..A000120(n)} 2^k_i;
G.f. Sum_{n>=0} s_n(x)*z^n = Product_{k>=0} (1 + (x^(2^k)+1)*z^(2^k)) (0Let x>1, t>0 be real numbers. Then
Sum_{n>=0} 1/s_n(x)^t = Product_{k>=0} (1 + 1/(x^(2^k)+1)^t);
Sum_{n>=0} (-1)^A000120(n)/s_n(x)^t = Product_{k>=0} (1 - 1/(x^(2^k)+1)^t).
In particular, for t=1, x>1, we have
Sum_{n>=0} (-1)^A000120(n)/s_n(x) = 1 - 1/x. (End)
From Valentin Bakoev, Jul 11 2020: (Start)
(See my comment about the matrix M_n.) Denote by T(i,j) the number in the i-th row and j-th column of M_n (0 <= i, j < 2^n). When i>=j, T(i,j) is the j-th number in the i-th row of the Sierpinski's triangle. For given i and j, we denote by k the largest integer of the type k=2^m and kT(i,0) = T(i,i) = 1, or
T(i,j) = 0 if i < j, or
T(i,j) = T(i-k,j), if j < k, or
T(i,j) = T(i-k,j-k), if j >= k.
Thus, for given i and j, T(i,j) can be computed in O(log_2(i)) steps. (End)

Additional links from Lekraj Beedassy, Jan 22 2004

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

A255483 Infinite square array read by antidiagonals downwards: T(0,m) = prime(m), m >= 1; for n >= 1, T(n,m) = T(n-1,m)*T(n-1,m+1)/gcd(T(n-1,m), T(n-1,m+1))^2, m >= 1.

Original entry on oeis.org

2, 3, 6, 5, 15, 10, 7, 35, 21, 210, 11, 77, 55, 1155, 22, 13, 143, 91, 5005, 39, 858, 17, 221, 187, 17017, 85, 3315, 1870, 19, 323, 247, 46189, 133, 11305, 5187, 9699690, 23, 437, 391, 96577, 253, 33649, 21505, 111546435, 46

Offset: 0

N. J. A. Sloane, Feb 28 2015

The first column of the array is given by A123098; subsequent columns are obtained by applying the function A003961, i.e., replacing each prime factor by the next larger prime. - M. F. Hasler, Sep 17 2016
Interpretation with respect to A329329 from Peter Munn, Feb 08 2020: (Start)
With respect to the ring defined by A329329 and A059897, the first row gives powers of 3, the first column gives powers of 6, both in order of increasing exponent, and the body of the table gives their products. A329049 is the equivalent table in which the first column gives powers of 4.
A099884 is the equivalent table for the ring defined by A048720 and A003987. That ring is an image of the polynomial ring GF(2)[x] using a standard representation of the polynomials as integers. A329329 describes a comparable mapping to integers from the related polynomial ring GF(2)[x,y].
Using these mappings, the tables here and in A099884 are matching images: the first row represents powers of x, the first column represents powers of (x+1) and the body of the table gives their products.
Hugo van der Sanden's formula (see formula section) indicates that A019565 provides a mapping from A099884. In the wider terms described above, A019565 is an injective homomorphism between images of the 2 polynomial rings, and maps the image of each GF(2)[x] polynomial to the image of the equivalent GF(2)[x,y] polynomial.
(End)

			The top left corner of the array, row index 0..5, column index 1..10:
    2,    3,     5,     7,    11,     13,     17,     19,      23,      29
    6,   15,    35,    77,   143,    221,    323,    437,     667,     899
   10,   21,    55,    91,   187,    247,    391,    551,     713,    1073
  210, 1155,  5005, 17017, 46189,  96577, 215441, 392863,  765049, 1363783
   22,   39,    85,   133,   253,    377,    527,    703,     943,    1247
  858, 3315, 11305, 33649, 95381, 198679, 370481, 662929, 1175921, 1816879

Alois P. Heinz, Antidiagonals n = 0..125, flattened
C. Cobeli, A. Zaharescu, A game with divisors and absolute differences of exponents, Journal of Difference Equations and Applications, Vol. 20, #11, 2014.
C. Cobeli, A. Zaharescu, A game with divisors and absolute differences of exponents, arXiv:1411.1334 [math.NT], 2014.
Discussion of SeqFan-mailing list
Index entries for sequences related to Gilbreath conjecture and transform

First two columns = A123098, A276804.
Rows = A000040, A006094, A090076, A046302, ...
A kind of generalization of A036262.
Cf. A003961, A007913, A019565, A048675, A066117, A099884.
Transpose: A276578, terms sorted into ascending order: A276579.
A003987, A048720, A059897, A329049 relate to the A329329 polynomial ring interpretation.

T:= proc(n, m) option remember; `if`(n=0, ithprime(m),
      T(n-1, m)*T(n-1, m+1)/igcd(T(n-1, m), T(n-1, m+1))^2)
    end:
seq(seq(T(n, 1+d-n), n=0..d), d=0..10);  # Alois P. Heinz, Feb 28 2015

T[n_, m_] := T[n, m] = If[n == 0, Prime[m], T[n-1, m]*T[n-1, m+1]/GCD[T[n-1, m], T[n-1, m+1]]^2]; Table[Table[T[n, 1+d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

T=matrix(N=15,N);for(j=1,N,T[1,j]=prime(j));(f(x,y)=x*y/gcd(x,y)^2);for(k=1,N-1,for(j=1,N-k,T[k+1,j]=f(T[k,j],T[k,j+1])));A255483=concat(vector(N,i,vector(i,j,T[j,1+i-j]))) \\ M. F. Hasler, Sep 17 2016

A255483(n,k)=prod(j=0,n,if(bitand(n-j,j),1,prime(j+k))) \\ M. F. Hasler, Sep 18 2016

(define (A255483 n) (A255483bi (A002262 n) (+ 1 (A025581 n))))
;; Then use either an almost standalone version (requiring only A000040):
(define (A255483bi row col) (if (zero? row) (A000040 col) (let ((a (A255483bi (- row 1) col)) (b (A255483bi (- row 1) (+ col 1)))) (/ (lcm a b) (gcd a b)))))
;; Or one based on M. F. Hasler's new recurrence:
(define (A255483bi row col) (if (= 1 col) (A123098 row) (A003961 (A255483bi row (- col 1)))))
;; Antti Karttunen, Sep 18 2016

T(n,1) = A123098(n), T(n,m+1) = A003961(T(n,m)), for all n >= 0, m >= 1. - M. F. Hasler, Sep 17 2016
T(n,m) = Prod_{k=0..n} prime(k+m)^(!(n-k & k)) where !x is 1 if x=0 and 0 else, and & is binary AND. - M. F. Hasler, Sep 18 2016
From Antti Karttunen, Sep 18 2016: (Start)
For n >= 1, m >= 1, T(n,m) = lcm(T(n-1,m),T(n-1,m+1)) / gcd(T(n-1,m),T(n-1,m+1)).
T(n,k) = A007913(A066117(n+1,k)).
T(n,k) = A019565(A099884(n,k-1)) [After Hugo van der Sanden's observations on SeqFan-list].
(End)
From Peter Munn, Jan 08 2020: (Start)
T(0,1) = 2, and for n >= 0, k >= 1, T(n+1,k) = A329329(T(n,k), 6), T(n,k+1) = A329329(T(n,k), 3).
T(n,k) = A329329(T(n,1), T(0,k)).
(End)

A277810 Square array A(r,c) = A019565(A277820(r,c)), read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

2, 6, 3, 10, 15, 30, 210, 21, 14, 5, 22, 1155, 462, 35, 70, 858, 39, 910, 55, 330, 105, 1870, 3315, 72930, 5005, 2002, 33, 42, 9699690, 5187, 2926, 85, 714, 2145, 770, 7, 46, 111546435, 238602, 11305, 248710, 3927, 390, 77, 154, 4002, 87, 93763670, 21505, 152490, 440895, 3094, 91, 546, 231, 7130, 13485, 620310, 1078282205, 2306486, 9867, 114114, 17017, 170170, 1365, 2310

Offset: 1

Antti Karttunen, Nov 01 2016

Permutation of squarefree numbers (A005117) after their initial term 1.

			The top left corner of the array:
    2,    6,     10,   210,      22,    858,      1870,    9699690
    3,   15,     21,  1155,      39,   3315,      5187,  111546435
   30,   14,    462,   910,   72930,   2926,    238602,   93763670
    5,   35,     55,  5005,      85,  11305,     21505, 1078282205
   70,  330,   2002,   714,  248710, 152490,   2306486,   60138078
  105,   33,   2145,  3927,  440895,   9867,   1870935,  691587897
   42,  770,    390,  3094,  114114, 520030,    162690,  581334754
    7,   77,     91, 17017,     133,  33649,     50141, 6685349671
  154,  546, 170170,   570,    6118, 254562, 357505330,   51269790
  231, 1365,   7293,  3135, 1312311, 983535,  11599797,  589602585

Antti Karttunen, Table of n, a(n) for n = 1..1540; the first 55 antidiagonals of the array

Transpose: A277809.
The topmost row: A123098, the leftmost column: A277811.
Cf. A005117, A019565, A277820.

(define (A277810 n) (A277810bi (A002260 n) (A004736 n)))
(define (A277810bi row col) (A019565 (A277820bi row col)))

A(r,c) = A019565(A277820(r,c)).

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

A334205 Under the isomorphism defined in A329329, of polynomials in GF(2)[x,y] to positive integers, a(n) is the image of the polynomial that results when x+1 is substituted for x in the polynomial with image n.

Original entry on oeis.org

1, 2, 6, 4, 10, 3, 210, 8, 36, 5, 22, 24, 858, 105, 15, 16, 1870, 72, 9699690, 40, 35, 11, 46, 12, 100, 429, 216, 840, 4002, 30, 7130, 32, 33, 935, 21, 9, 160660290, 4849845, 143, 20, 20746, 70, 1008940218, 88, 360, 23, 2569288370, 96, 44100, 200, 2805, 3432, 32589158477190044730, 108, 55, 420, 1616615, 2001, 118, 60, 21594, 3565

Offset: 1

Antti Karttunen and Peter Munn, May 04 2020

nonn

Under the isomorphism (defined in A329329), A059897(.,.), A329329(.,.) and A003961(.) represent polynomial addition, multiplication and multiplication by x respectively; prime(i+1) represents the polynomial x^i.
The equivalent sequence with y+1 substituted for y is A268385.
Self-inverse permutation of natural numbers. Squarefree numbers are mapped to squarefree numbers, squares are mapped to squares, and in general the sequence permutes {m : A267116(m) = k} for any k.
From Peter Munn, May 31 2020: (Start)
The odd numbers represent the polynomials that have x as a factor. So the odd bisection's terms represent polynomials with (x+1) as a factor. They are a permutation of A268390.
A193231 is an equivalent sequence with respect to GF(2)[x]. See the formula showing A019565 as the related injective homomorphism, mapping the usual encoding of GF(2) polynomials in x to their equivalent A329329-defined representation.
(End)

			Calculation for n = 5. 5 = prime(3) = prime(2+1) is the image of the polynomial x^2. Substituting x+1 for x, this becomes (x+1)^2 = x^2 + (1+1)x + 1 = x^2 + 1, as 1 + 1 = 0 in GF(2). The image of x^2 + 1 is A059897(prime(3), prime(1)) = A059897(5, 2) = 10. So a(5) = 10. (Note that A059897 gives the same result as multiplication when its operands are different terms of A050376, such as prime numbers.)

Antti Karttunen, Table of n, a(n) for n = 1..3670
Wikipedia, Polynomial ring
Index entries for sequences that are permutations of the natural numbers

Cf. A123098, A267116.
Equivalent GF(2)[x] sequence is A193231 (via A019565).
Equivalent sequences for other substitutions: x -> 0: A006519, (x -> y, y -> x): A225546, y -> y+1: A268385, x -> x^2: A319525.
Cf. A268390 (ordered odd bisection).
A003961, A007913, A008833, A059897, A329329 are used to express relationship between terms of this sequence.

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A225546(n) = if(1==n,1,my(f=factor(n),u=#binary(vecmax(f[, 2])),prods=vector(u,x,1),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),prods[i] *= f[k,1])); m<<=1); prod(i=1,u,prime(i)^A048675(prods[i])));
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) }; \\ From A193231
A268385(n) = if(1==n, n, my(f=factor(n)); prod(i=1,#f~,f[i,1]^A193231(f[i,2])));
A334205(n) = A225546(A268385(A225546(n)));

\\ This program is better for larger values. A048675 and A193231 as in above:
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A334205(n) = if(1==n, n, if(issquare(n), A334205(sqrtint(n))^2, A019565(A193231(A048675(core(n)))) * A334205(n/core(n)))); \\ Antti Karttunen, May 24 2020

a(prime(i)^j) = A123098(i-1)^j, a(A059897(n, k)) = A059897(a(n), a(k)).
a(n) = A225546(A268385(A225546(n))).
a(A003961(n)) = A059897(a(n), A003961(a(n))) = A329329(6, a(n)).
a(n^2) = a(n)^2.
a(n) = a(A007913(n)) * a(A008833(n)).
a(A019565(n)) = A019565(A193231(n)).
A267116(a(n)) = A267116(n).

A255484 a(n) = Product_{k=0..n} prime(k+1)*(binomial(n,k) mod 2).

Original entry on oeis.org

2, 6, 0, 210, 0, 0, 0, 9699690, 0, 0, 0, 0, 0, 0, 0, 32589158477190044730, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 525896479052627740771371797072411912900610967452630, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Feb 28 2015

nonn

A123098 is a much better version of this sequence.

Chai Wah Wu, Table of n, a(n) for n = 0..510

Cf. A123098, A255483.

f:=n->mul(ithprime(k+1)*(binomial(n,k) mod 2),k=0..n);
[seq(f(n),n=0..60)];

from operator import mul
from functools import reduce
from sympy import prime
def A255484(n):
    return reduce(mul,(0 if ~n & k else prime(k+1) for k in range(n+1))) # Chai Wah Wu, Feb 09 2016

A277809 Transpose of square array A277810.

Original entry on oeis.org

2, 3, 6, 30, 15, 10, 5, 14, 21, 210, 70, 35, 462, 1155, 22, 105, 330, 55, 910, 39, 858, 42, 33, 2002, 5005, 72930, 3315, 1870, 7, 770, 2145, 714, 85, 2926, 5187, 9699690, 154, 77, 390, 3927, 248710, 11305, 238602, 111546435, 46, 231, 546, 91, 3094, 440895, 152490, 21505, 93763670, 87, 4002, 2310, 1365, 170170, 17017, 114114, 9867, 2306486, 1078282205, 620310, 13485, 7130

Offset: 1

Antti Karttunen, Nov 01 2016

See A277810.

			The top left corner of the array:
     2,    3,     30,     5,      70,     105,     42,     7,       154
     6,   15,     14,    35,     330,      33,    770,    77,       546
    10,   21,    462,    55,    2002,    2145,    390,    91,    170170
   210, 1155,    910,  5005,     714,    3927,   3094, 17017,       570
    22,   39,  72930,    85,  248710,  440895, 114114,   133,      6118
   858, 3315,   2926, 11305,  152490,    9867, 520030, 33649,    254562
  1870, 5187, 238602, 21505, 2306486, 1870935, 162690, 50141, 357505330

Antti Karttunen, Table of n, a(n) for n = 1..1275; the first 50 antidiagonals of the array

Transpose: A277810.
The topmost row: A277811, the leftmost column: A123098.
Permutation of squarefree numbers (A005117) after their initial term 1.
Cf. A019565, A277819.

(define (A277809 n) (A277810bi (A004736 n) (A002260 n))) ;; Code for A277810bi given in A277810.

A(r,c) = A019565(A277819(r,c)).

Showing 1-8 of 8 results.

cp's OEIS Frontend

A047999 Sierpiński's [Sierpinski's] triangle (or gasket): triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 2.

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

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A255483 Infinite square array read by antidiagonals downwards: T(0,m) = prime(m), m >= 1; for n >= 1, T(n,m) = T(n-1,m)*T(n-1,m+1)/gcd(T(n-1,m), T(n-1,m+1))^2, m >= 1.

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A277810 Square array A(r,c) = A019565(A277820(r,c)), read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

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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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A334205 Under the isomorphism defined in A329329, of polynomials in GF(2)[x,y] to positive integers, a(n) is the image of the polynomial that results when x+1 is substituted for x in the polynomial with image n.

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A255484 a(n) = Product_{k=0..n} prime(k+1)*(binomial(n,k) mod 2).