OFFSET
0,2
FORMULA
a(n) = 2^(n + 1) * 5^n * (2^(n + 1) * HurwitzZeta(-n, 1/20) - HurwitzZeta(-n, 1/10)).
a(n) = Im(p(n)) where p(n) = 2*i*(1 + Sum_{j=0..n} binomial(n, j)*polylog(-j, i)*5^j.
a(n) = (1/(5*(n+1))) * Sum_{j=0..n+1} Bernoulli(j, 0) * binomial(n+1, j) * (10^j - 20^j).
MAPLE
a := n -> local j; add(binomial(n, j)*euler(j, 0)*10^j, j = 0..n):
# Alternative:
a := n -> add(10^j*(1-2^j)*bernoulli(j)*binomial(n+1, j), j = 0..n+1) / (5*(n+1)):
seq(a(n), n = 0..18);
MATHEMATICA
a[n_] := 5^n (4^(n+1) HurwitzZeta[-n, 1/(20)] - 2^(n + 1) HurwitzZeta[-n, 1/(10)]);
Table[Round[N[a[n], 64]], {n, 0, 18}]
PROG
(SageMath)
from mpmath import *
mp.dps = 32; mp.pretty = True
def a(n): return int(imag(2*I*(1+sum(binomial(n, j)*polylog(-j, I)*5^j for j in range(n+1)))))
print([a(n) for n in range(19)])
CROSSREFS
KEYWORD
sign
AUTHOR
Peter Luschny, Nov 13 2024
STATUS
approved