A377665 a(n) = Sum_{j=0..n} binomial(n, j) * Euler(j, 0) * 10^j. Row 5 of A377666.

1, -4, -9, 236, 981, -47524, -295029, 20208716, 167213961, -14741279044, -152462570049, 16429489441196, 203906790454941, -25968596099278564, -376012858170009069, 55254540434093713676, 914353480122881739921, -152277985980992039230084, -2834887281233334168196089

Offset: 0

Author

Peter Luschny, Nov 13 2024

Keywords

• sign

Crossrefs

Cf. A377666.

Programs

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}]

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)])

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).