cp's OEIS Frontend

This is to A047999 "Triangle formed by reading Pascal's triangle mod 2" as A007188 "Multiplicative encoding of Pascal triangle: Product p(i+1)^C(n,i)" is to A007318 "Pascal's triangle read by rows." a(2^n - 1) = primorial(2^n) = A002110(A000079(n)). In row(n) the primes with exponent 1 form row(n) of a Sierpinski sieve, so this sequence is a kind of Gödelization of a Sierpinski sieve.

All terms are divisible by 2 and the n-th term, a(n-1), is also divisible by prime(n). This sequence appears as first column of the square array A255483; its second column A276804 is very similar, with all prime factors shifted to the net larger prime (cf. A003961). - M. F. Hasler, Sep 17 2016

a(n) is the n-th power of 6 in the ring defined in A329329. - Peter Munn, Jan 04 2020

Seen as a triangle read by rows: T(n,k) = A000040(n), 1 <= k <= n; row sums = A033286; central terms = A031368. - Reinhard Zumkeller, Aug 05 2009

Seen as a square array read by antidiagonals, a subtable of the binary operation multiplication tables A297845, A306697 and A329329. - Peter Munn, Jan 15 2020

Multiplicative with a(p^e) = p^A048724(e), where A048724(e) = (e XOR 2e).

Multiples of 8 in the ring defined in A329329. - Peter Munn, Jan 17 2020

The ones in the binary expansion of n encode the Fermi-Dirac factors of a(n).

The following table gives the rank of the bit corresponding to the Fermi-Dirac factor p^2^k:

...

7| 9

5| 5 8

3| 2 4 7

2| 0 1 3 6

---+--------

p/k| 0 1 2 3 ...

This sequence is a bijection from the nonnegative integers to the positive integers with inverse A344536.

This sequence establishes a bijection from A261195 to A225547.

This sequence and A344535 each map between two useful choices for encoding sets of elements drawn from a 2-dimensional array. To give a very specific example, each mapping is an isomorphism between two alternative integer representations of the polynomial ring GF2[x,y]. The relevant set is {x^i*y^j : i, j >= 0}. The mappings between the two representations of the ring's addition operation are from XOR (A003987) to A059897(.,.) and for the multiplication operation, they are from A329331(.,.) to A329329(.,.). - Peter Munn, May 31 2021

When creating A329329, the author realized it was isomorphic to multiplication in the GF(2)[x,y] polynomial ring. However, A329329 was unusual in having A059897(.,.) as additive operator, whereas the equivalent univariate polynomial ring, GF(2)[x], is more commonly mapped (to integers) with bitwise exclusive-or (A003987) representing polynomial addition (and A048720(.,.) representing polynomial multiplication). This sequence shows how multiplication in GF(2)[x] can look when mapped to integers with A059897(.,.) representing polynomial addition.

The group defined by the binary operation A059897(.,.) over the positive integers is commutative with all elements self-inverse, and isomorphic to the additive group of the polynomial ring GF(2)[x]. There is a unique isomorphism extending each bijective mapping between respective minimal generating sets. The lexicographically earliest minimal generating set for the A059897 group is A050376, often called the Fermi-Dirac primes. The most meaningful generating set for the additive group of GF(2)[x] is {x^i: i >= 0}.

Using f to denote the intended isomorphism from GF(2)[x], we specify f(x^i) = A050376(i+1). This maps minimal generating sets of the additive groups, so the definition of f is completed by specifying f(a+b) = A059897(f(a), f(b)). We then calculate the image under f of polynomial multiplication in GF(2)[x], giving us this sequence as the matching multiplication operation for an isomorphic ring over the positive integers. Using g to denote the inverse of f, A(n,k) = f(g(n) * g(k)).

Note that A050376 is closed with respect to A(.,.).

Recall that GF(2)[x] is more usually mapped to integers with A003987(.,.) as addition and A048720(.,.) as multiplication. With this usual mapping, under which A000079(i) is the image of x^i, A052330(.) is the relevant isomorphism from nonnegative integers under A048720(.,.) and A003987(.,.) to positive integers under A(.,.) and A059897(.,.), with A052331(.) its inverse.

This sequence defines a multiplication operation that goes with bitwise exclusive-or (A003987) as addition operation to form a ring over the nonnegative integers. It is isomorphic to the polynomial ring GF(2)[x,y], as is the ring defined in A329329.

The ring defined by A329329 is unusual in that it has A059897(.,.) as its addition operation, given that A059897 has more similarities to integer multiplication. A003987, which is isomorphic to A059897 as a binary operation, seems a more standard choice for an addition operator.

However, as explained in A329329, A059897 has a natural choice for mapping a generating set to the 2-dimensions (x and y) of the generating set for the additive group of GF(2)[x,y]. Instead, A003987 needs a pairing function to map its most natural generating set {2^k: k >= 0} onto {x^i * y^j: i >= 0, j >= 0}.

The choice made here was to map 2^k onto the 2 dimensions of x^i * y^j, by proceeding through x and y dimensions as when reading an array by antidiagonals. 2^0 = 1 is mapped to (x^0 * y*0) = 1, 2^1 = 2 is mapped to (x^1 * y^0) = x, 2^2 = 4 to (x^0 * y^1) = y, 8 to (x^2 * y^0) = x^2, and so on, 16 mapped to xy, 32 to y^2, 64 to x^3, etc. With this mapping, it can be shown that the result of the multiplying the polynomial images of 2^i and 2^j is the image of 2^A054237(i,j).