

A247209


Number of terms in generalized SwinnertonDyer polynomials.


1



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

0,2


COMMENTS

If the sequence of primes used in the construction of SwinnertonDyer 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(n1) 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. 121136.
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, SwinnertonDyer Polynomial.


EXAMPLE

a(3) = 35. For the three numbers a, b, c, the general SwinnertonDyer 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^44a^3b+6a^2b^24ab^3+b^44a^3c+4a^2bc+4ab^2c4b^3c+6a^2c^2+4abc^2+6b^2c^24ac^34bc^3+c^4 4a^3z^2+4a^2bz^2+4ab^2z^24b^3z^2+4a^2cz^240abcz^2+ 4b^2cz^2+4ac^2z^2+4bc^2z^24c^3z^2+6a^2z^4+4abz^4+ 6b^2z^4+4acz^4+4bcz^4+6c^2z^44az^64bz^64cz^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



