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A338862 a(n) is the number of polynomials of degree 2*n over the field GF(2) that have no factors of odd degree. 0
1, 1, 4, 13, 49, 175, 655, 2437, 9208, 34867, 132952, 508621, 1953580, 7524625, 29061835, 112493680, 436330753, 1695388480, 6598016866, 25714222228, 100343852938, 392023844362, 1533182752336, 6001993189687, 23517048084424, 92220047277892, 361906295452669, 1421252193947311 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) is the number of partitions of n into parts k of A001037(2*k) sorts. - Joerg Arndt, Nov 13 2020

LINKS

Table of n, a(n) for n=0..27.

A. Swaminathan, How many monic polynomials modulo 2 have an odd-degree factor?, Mathematics StackExchange.

FORMULA

G.f.: 1 / ( Product_{k>=1} (1-x^k)^A001037(2*k) ). - Joerg Arndt, Nov 13 2020

EXAMPLE

For n=2 the a(2) = 4 polynomials are X^4 + X + 1, X^4 + X^2 + 1, X^4 + X^3 + 1, and X^4 + X^3 + X^2 + X + 1.  Of these X^4+X^2+1 = (X^2+X+1)^2 while the others are irreducible over GF(2).

MAPLE

f:= proc(d) local P, x, X, count, L, F;

  count:= 0;

  for x from 2^d+1 to 2^(d+1)-1 by 2 do

  L:= convert(x, base, 2);

  if convert(L, `+`)::even then next fi;

  P:= add(L[i+1]*X^i, i=0..d);

  F:= map(t -> t[1], (Factors(P) mod 2)[2]);

  if andmap(t -> degree(t)::even, F) then count:= count+1 fi

  od:

count

end proc:

seq(f(i), i=2..20, 2);

PROG

(PARI) b(n)=sumdiv(n, d, moebius(d)*2^(n/d))/n; \\ A001037

N=33;  x='x+O('x^N);

Vec( 1 / prod(k=1, N, (1-x^k)^b(2*k) ) )

\\ Joerg Arndt, Nov 13 2020

CROSSREFS

Sequence in context: A294298 A300579 A180007 * A097948 A096971 A149451

Adjacent sequences:  A338859 A338860 A338861 * A338863 A338864 A338865

KEYWORD

nonn

AUTHOR

Robert Israel, Nov 12 2020

STATUS

approved

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Last modified April 14 20:07 EDT 2021. Contains 342962 sequences. (Running on oeis4.)