A239035 Product of 8 consecutive integers. a(n) = RisingFactorial(n, 8).

0, 40320, 362880, 1814400, 6652800, 19958400, 51891840, 121080960, 259459200, 518918400, 980179200, 1764322560, 3047466240, 5079110400, 8204716800, 12893126400, 19769460480, 29654190720, 43609104000, 62990928000, 89513424000, 125318793600, 173059286400

Offset: 0

Author

Michel Marcus, Mar 09 2014

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Vincent Thill, Nombres consécutifs et carrés, author's blog, Aug 03 2013 (consecutive numbers and squares).
• Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Crossrefs

Cf. Product of n consecutive integers: A002378 (n=2), A007531 (n=3), A052762 (n=4), A052787 (n=5), A053625 (n=6), A159083 (n=7).

Programs

Magma

[(n^4+14*n^3+63*n^2+98*n+28)^2-(28+8*n)^2: n in [0..30]]; // Vincenzo Librandi, Mar 11 2014

Mathematica

CoefficientList[Series[40320 x/(1 - x)^9, {x, 0, 30}], x] (* Vincenzo Librandi, Mar 11 2014 *)

PARI

a(n) = prod(k=0, 7, n+k);

PARI

concat([0], Vec(40320*x/(1-x)^9  + O(x^100))) \\ Colin Barker, Mar 09 2014

Sage

print([rising_factorial(n, 8) for n in range(23)]) # Peter Luschny, Mar 22 2022

Formula

a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7).

a(n) = (n^4+14*n^3+63*n^2+98*n+28)^2 - (28+8*n)^2. (see Thill link).

G.f.: 40320*x / (1-x)^9. - Colin Barker, Mar 09 2014

From Amiram Eldar, Mar 08 2022: (Start)

Sum_{n>=1} 1/a(n) = 1/35280.

Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(2)/315 - 1163/66150. (End)