

A346515


a(n) = n*(n+2)*(n+4)*(n+6)*(n+8)*(n+10)*(n+12)*(n+14).


1



0, 2027025, 10321920, 34459425, 92897280, 218243025, 464486400, 916620705, 1703116800, 3011753745, 5109350400, 8365982625, 13284311040, 20534684625, 30996725760, 45808142625, 66421555200, 94670161425, 132843110400, 183771489825, 250925875200, 338526428625, 451666575360
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OFFSET

0,2


COMMENTS

a(n) can always be expressed as the difference of two squares: x^2  y^2.
A346514(n) gives the xvalues for each product. The yvalues being A152691(n+7).
More generally, for any k, we have: n*(n+k)*(n+2*k)*...*(n+7*k) = a(n,k) = x(n,k)^2  y(n,k)^2, where
x(n,k) = n^4 + 14*k*n^3 + 63*k^2*n^2 + 98*k^3*n + 28*k^4,
y(n,k) = 8*k^3*n + 28*k^4.
A239035(n) corresponds to a(n,k) in the case k = 1, with related y(n,k) = A346376(n).


LINKS

Table of n, a(n) for n=0..22.
Index entries for linear recurrences with constant coefficients, signature (9,36,84,126,126,84,36,9,1).


FORMULA

a(n) = A346514(n)^2  A152691(n+7)^2.


MATHEMATICA

a[n_] := (n + 14)!!/(n  2)!!; Array[a, 23, 0] (* Amiram Eldar, Jul 22 2021 *)


CROSSREFS

Cf. A239035, A190577, A346514, A346376.
Sequence in context: A032754 A271768 A104441 * A290037 A289956 A263892
Adjacent sequences: A346512 A346513 A346514 * A346516 A346517 A346518


KEYWORD

nonn,easy


AUTHOR

Lamine Ngom, Jul 21 2021


STATUS

approved



