A346515 a(n) = n*(n+2)*(n+4)*(n+6)*(n+8)*(n+10)*(n+12)*(n+14).

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

Offset

0,2

Comments

a(n) can always be expressed as the difference of two squares: x^2 - y^2.

A346514(n) gives the x-values for each product. The y-values 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