OFFSET
1,1
COMMENTS
A relation between fourth powers and the sum of fifth and seventh powers. See the first formula, which is from Beiler.
REFERENCES
Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 161.
LINKS
Temple Rice Hollcroft, On sums of powers of n consecutive integers, Bulletin of the American Mathematical Society 59 (1953), nr. 6, p. 526 (574t).
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).
FORMULA
a(n) = 2*(Sum_{k=1..n} k)^4 = Sum_{k=1..n} (k^5 + k^7).
a(n) = 2*A059977(n-1).
G.f.: -2*x*(1+72*x+603*x^2+1168*x^3+603*x^4+72*x^5+x^6) / (x-1)^9. - R. J. Mathar, Dec 13 2011
a(n) = 2*(A000217(n)^4). - Zak Seidov, Jan 21 2012
From Amiram Eldar, Apr 09 2024: (Start)
Sum_{n>=1} 1/a(n) = 8*Pi^4/45 + 80*Pi^2/3 - 280.
Sum_{n>=1} (-1)^(n+1)/a(n) = 280 - 320*log(2) - 48*zeta(3). (End)
MAPLE
MATHEMATICA
Table[n^4 (n+1)^4/8, {n, 100}] (* Wesley Ivan Hurt, Nov 12 2013 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Martin Renner, Dec 11 2011
STATUS
approved