OFFSET
0,1
COMMENTS
The product of eight positive integers shifted by 2; i.e., m * (m+2) * (m+4) * ... * (m+14) = A346515(m) can always be expressed as the difference of two squares: x^2 - y^2.
This sequence gives the x-values for each product. The y-values are 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) = 4*k^3*(2*n + 7*k).
This sequence is y(n,k) in the case k = 2, with related y(n,k) = A152691(n+7).
LINKS
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
G.f.: (448 - 727*x + 179*x^2 + 235*x^3 - 111*x^4)/(1 - x)^5. - Stefano Spezia, Jul 22 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Lamine Ngom, Jul 21 2021
STATUS
approved