A247209 Number of terms in generalized Swinnerton-Dyer polynomials.

1, 2, 6, 35, 495, 20349, 2760681, 1329890705, 2353351951665, 15481400876017505, 379554034822178909121, 34676179189150610052785025, 11806724418359403847522843860225, 14998128029851443976142151169687652865, 71221988684076361563783957084457295633613825

Offset: 0

Author

Morgan L. Owens, Nov 25 2014

Keywords

• nonn
• easy

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

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..60
• 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.

Crossrefs

Cf. A153731.

Programs

Maple

a:= n-> (t-> binomial(t+n, t))(2^(n-1)):
seq(a(n), n=0..14);  # Alois P. Heinz, Nov 28 2024

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]

Extensions

a(0) = 1 prepended by Michel Marcus, Jun 12 2022