A048631 Xfactorials - like factorials but use carryless GF(2)[ X ] polynomial multiplication.

1, 1, 2, 6, 24, 120, 272, 1904, 15232, 124800, 848640, 7507200, 39738368, 433441792, 2589116416, 30419859456, 486717751296, 8128101580800, 132557598294016, 1971862458400768, 30421253686034432, 512675443057623040, 7176891455747129344, 130521457800367308800

Offset

0,3

Comments

In formula X stands for the multiplication in a ring of GF(2)[ X ] polynomials.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..475

Vaclav Kotesovec, Graph a(n+1)/a(n)

Formula

a(0) = 1, a(n) = n X a(n-1) (see the Maple function Xfactorial given below).

Using the notations introduced in A355891, we have a(n) = ivgenpoly(Product_{i=1..n} genpoly(n)). As an example, n = 6 corresponds to 1*x*(x+1)*x^2*(x^2+1)*(x^2+x) = x^8+x^4 in GF(2)[x], so a(6) = 2^8 + 2^4 = 272. - Jianing Song, Sep 30 2022

Maple

Xfactorial := proc(n) option remember; if n=0 then 1
                else Xmult(n, Xfactorial(n-1)) fi
              end:
Xmult := proc(n, m) option remember; if n=0 then 0
           else Bits[Xor](((n mod 2)*m), Xmult(floor(n/2), m*2)) fi
         end:
seq(Xfactorial(n), n=0..23);

Mathematica

Xmult[nn_, mm_] := Module[{n = nn, m = mm, s = 0}, While[n > 0, If[1 == Mod[n, 2], s = BitXor[s, m]]; n = Floor[n/2]; m = m*2]; s];
Xfactorial[n_] := Xfactorial[n] = If[0 == n, 1, Xmult[n, Xfactorial[n - 1]] ];
Table[Xfactorial[n], {n, 0, 21}] (* Jean-François Alcover, Mar 04 2016, updated Mar 06 2016 after Maple *)

Prog

(PARI) a(n)=my(s=Mod(1, 2)); for(k=1, n, s*=Pol(binary(k))); fromdigits(Vec(lift(s)), 2) \\ Charles R Greathouse IV, Oct 03 2016

Crossrefs

Cf. A000142, A048720, A048632, A061922.

Sequence in context: A083267 A130480 A000804 * A263700 A263701 A263702

Adjacent sequences: A048628 A048629 A048630 * A048632 A048633 A048634

Keyword

easy,nonn

Author

Antti Karttunen, Jul 14 1999

Status

approved