OFFSET
0,2
COMMENTS
If the sequence of primes used in the construction of Swinnerton-Dyer polynomials is replaced by the generic sequence a_1, a_2, ..., a_n, this sequence gives the number of terms in the resulting multivariate polynomial (treating the a_n as variables).
a(n-1) is the number of monomials obtained when multiplying all the possible cases Sum_{k=1..n} e_k*sqrt(x_k) where e_1 is 1 and all other e_i are +1 or -1; so that 1/(Sum_{k=1..n} sqrt(x_k)) is transformed into a fraction whose denominator has no radicals. See the French link. - Michel Marcus, Jun 12 2022
LINKS
Allan Berele and Stefan Catoiu, Rationalizing Denominators, Mathematics Magazine, Vol. 88, No. 2 (2015), pp. 121-136.
Les Tablettes du Chercheur, Problem 21, Solution to problem 21, Addition to problem 21, pp. 4, 30 and 64, 1892 (in French).
Eric Weisstein's World of Mathematics, Swinnerton-Dyer Polynomial.
EXAMPLE
a(3) = 35. For the three numbers a, b, c, the general Swinnerton-Dyer polynomial is
(sqrt(a)+sqrt(b)+sqrt(c)-z)(-sqrt(a)+sqrt(b)+sqrt(c)-z)(sqrt(a)-sqrt(b)+sqrt(c)-z)(-sqrt(a)-sqrt(b)+sqrt(c)-z)(sqrt(a)+sqrt(b)-sqrt(c)-z)(-sqrt(a)+sqrt(b)-sqrt(c)-z)(sqrt(a)-sqrt(b)-sqrt(c)-z)(-sqrt(a)-sqrt(b)-sqrt(c)-z)
which expands to
a^4-4a^3b+6a^2b^2-4ab^3+b^4-4a^3c+4a^2bc+4ab^2c-4b^3c+6a^2c^2+4abc^2+6b^2c^2-4ac^3-4bc^3+c^4- 4a^3z^2+4a^2bz^2+4ab^2z^2-4b^3z^2+4a^2cz^2-40abcz^2+ 4b^2cz^2+4ac^2z^2+4bc^2z^2-4c^3z^2+6a^2z^4+4abz^4+ 6b^2z^4+4acz^4+4bcz^4+6c^2z^4-4az^6-4bz^6-4cz^6+z^8
with 35 terms.
MATHEMATICA
a[n_]:= Module[{a, x}, Length@Fold[Expand[(#1 /. x -> x + #2) (#1 /. x -> x - #2)] &, x, Sqrt[a /@ Range[n]]]]; a[0] = 1; Array[a, 5, 0] (* or *)
a[n_]:= Binomial[2^(n - 1) + n, 2^(n - 1)]; Array[a, 10, 0]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Morgan L. Owens, Nov 25 2014
EXTENSIONS
a(0) = 1 prepended by Michel Marcus, Jun 12 2022
STATUS
approved