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A247209 Number of terms in generalized Swinnerton-Dyer polynomials. 1
1, 2, 6, 35, 495, 20349, 2760681, 1329890705, 2353351951665, 15481400876017505, 379554034822178909121, 34676179189150610052785025, 11806724418359403847522843860225, 14998128029851443976142151169687652865, 71221988684076361563783957084457295633613825 (list; graph; refs; listen; history; text; internal format)
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 mononials 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

Table of n, a(n) for n=0..14.

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

Cf. A153731.

Sequence in context: A334162 A173399 A193190 * A324577 A324582 A275600

Adjacent sequences: A247206 A247207 A247208 * A247210 A247211 A247212

KEYWORD

nonn,easy

AUTHOR

Morgan L. Owens, Nov 25 2014

EXTENSIONS

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

STATUS

approved

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Last modified December 1 23:44 EST 2022. Contains 358485 sequences. (Running on oeis4.)