A054777 a(n) = 4*n*(4*n-1)*(4*n-2)*(4*n-3).

0, 24, 1680, 11880, 43680, 116280, 255024, 491400, 863040, 1413720, 2193360, 3258024, 4669920, 6497400, 8814960, 11703240, 15249024, 19545240, 24690960, 30791400, 37957920, 46308024, 55965360, 67059720, 79727040, 94109400, 110355024, 128618280, 149059680, 171845880

Offset: 0

Author

Henry Bottomley, May 19 2000

Keywords

• easy
• nonn

References

• L. B. W. Jolley, Summation of Series, Dover, 1961.
• Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 268.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original german edition of "Theory and Application of Infinite Series")
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A052762, A060541, A104188.

Programs

Magma

[4*n*(4*n-1)*(4*n-2)*(4*n-3): n in [0..30]]; // Vincenzo Librandi, Oct 04 2011

Mathematica

a[n_] := 4*n*(4*n-1)*(4*n-2)*(4*n-3); Array[a, 40, 0] (* Amiram Eldar, Mar 08 2022 *)

Formula

a(n) = A052762(4n) = 24*A060541(n).

Sum_{n>=1} 1/a(n) = log(2)/4 - Pi/24 = 0.0423871012404116... [Jolley eq. 242] - Benoit Cloitre, Apr 05 2002

G.f. -24*x*(1 + 65*x + 155*x^2 + 35*x^3) / (x-1)^5. - R. J. Mathar, Oct 03 2011

Sum_{n>=1} (-1)^(n+1)/a(n) = log(sqrt(2)-1)/(6*sqrt(2)) - log(2)/24 + (1/(6*sqrt(2)) - 1/16)*Pi. - Amiram Eldar, Mar 08 2022