A247209 - OEIS

%I #44 Nov 28 2024 08:32:50

%S 1,2,6,35,495,20349,2760681,1329890705,2353351951665,

%T 15481400876017505,379554034822178909121,34676179189150610052785025,

%U 11806724418359403847522843860225,14998128029851443976142151169687652865,71221988684076361563783957084457295633613825

%N Number of terms in generalized Swinnerton-Dyer polynomials.

%C 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).

%C 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

%H Alois P. Heinz, <a href="/A247209/b247209.txt">Table of n, a(n) for n = 0..60</a>

%H Allan Berele and Stefan Catoiu, <a href="https://www.jstor.org/stable/10.4169/math.mag.88.2.121">Rationalizing Denominators</a>, Mathematics Magazine, Vol. 88, No. 2 (2015), pp. 121-136.

%H Les Tablettes du Chercheur, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k5816259c/f10.item">Problem 21</a>, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k5816259c/f36.item">Solution to problem 21</a>, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k5816259c/f70.item">Addition to problem 21</a>, pp. 4, 30 and 64, 1892 (in French).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Swinnerton-DyerPolynomial.html">Swinnerton-Dyer Polynomial</a>.

%e a(3) = 35. For the three numbers a, b, c, the general Swinnerton-Dyer polynomial is

%e (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)

%e which expands to

%e 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

%e with 35 terms.

%p a:= n-> (t-> binomial(t+n, t))(2^(n-1)):

%p seq(a(n), n=0..14);  # _Alois P. Heinz_, Nov 28 2024

%t 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 *)

%t a[n_]:= Binomial[2^(n - 1) + n, 2^(n - 1)]; Array[a, 10, 0]

%Y Cf. A153731.

%K nonn,easy

%O 0,2

%A _Morgan L. Owens_, Nov 25 2014

%E a(0) = 1 prepended by _Michel Marcus_, Jun 12 2022