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A263931
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a(n) = binomial(2*n, n) / Product(p prime | n < p <= 2*n).
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5
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1, 1, 2, 4, 2, 36, 12, 24, 90, 20, 4, 168, 28, 1400, 5400, 720, 90, 5940, 23100, 46200, 180180, 17160, 1560, 140400, 11700, 45864, 179928, 13328, 52360, 5969040, 397936, 795872, 3133746, 12345060, 726180, 2863224, 159068, 318136, 1255800, 4958800, 247940
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OFFSET
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0,3
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COMMENTS
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The highest exponent in the prime factorization of a(n) is A263922(n), n>=2.
a(n) is even for n>=2.
By the Erdős squarefree conjecture, proved in 1996, no a(n) with n >= 5 is squarefree. - Robert FERREOL, Sep 06 2022
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LINKS
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FORMULA
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MAPLE
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a := n -> binomial(2*n, n)/convert(select(isprime, {$n+1..2*n}), `*`):
seq(a(n), n=0..40);
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PROG
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(PARI) a(n) = { my(res = 1); forprime(p = 2, n, res*= p^(val(2*n, p) - 2*val(n, p))); forprime(p = n + 1, 2*n, res*= p^(val(2*n, p) - 2*val(n, p) - 1)); res } val(n, p) = my(r=0); while(n, r+=n\=p); r \\ David A. Corneth, Apr 03 2021
(Python)
from math import comb
from sympy import primorial
def A263931(n): return comb(m:=n<<1, n)*primorial(n, nth=False)//primorial(m, nth=False) if n else 1 # Chai Wah Wu, Sep 07 2022
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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