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A247209 Number of terms in generalized Swinnerton-Dyer polynomials. 0
2, 6, 35, 495, 20349, 2760681, 1329890705, 2353351951665, 15481400876017505, 379554034822178909121, 34676179189150610052785025, 11806724418359403847522843860225, 14998128029851443976142151169687652865, 71221988684076361563783957084457295633613825 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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).

LINKS

Table of n, a(n) for n=1..14.

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]]]]; Array[a, 5] (* or *)

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

CROSSREFS

Cf. A153731.

Sequence in context: A107868 A173399 A193190 * A324577 A324582 A275600

Adjacent sequences:  A247206 A247207 A247208 * A247210 A247211 A247212

KEYWORD

nonn,easy

AUTHOR

Morgan L. Owens, Nov 25 2014

STATUS

approved

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Last modified October 16 08:50 EDT 2019. Contains 328056 sequences. (Running on oeis4.)