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

%I A377665 #12 Mar 31 2025 01:44:58
%S A377665 1,-4,-9,236,981,-47524,-295029,20208716,167213961,-14741279044,
%T A377665 -152462570049,16429489441196,203906790454941,-25968596099278564,
%U A377665 -376012858170009069,55254540434093713676,914353480122881739921,-152277985980992039230084,-2834887281233334168196089
%N A377665 a(n) = Sum_{j=0..n} binomial(n, j) * Euler(j, 0) * 10^j. Row 5 of A377666.
%F A377665 a(n) = 2^(n + 1) * 5^n * (2^(n + 1) * HurwitzZeta(-n, 1/20) - HurwitzZeta(-n, 1/10)).
%F A377665 a(n) = Im(p(n)) where p(n) = 2*i*(1 + Sum_{j=0..n} binomial(n, j)*polylog(-j, i)*5^j).
%F A377665 a(n) = (1/(5*(n+1))) * Sum_{j=0..n+1} Bernoulli(j, 0) * binomial(n+1, j) * (10^j - 20^j).
%p A377665 a := n -> local j; add(binomial(n, j)*euler(j, 0)*10^j, j = 0..n):
%p A377665 # Alternative:
%p A377665 a := n -> add(10^j*(1-2^j)*bernoulli(j)*binomial(n+1, j), j = 0..n+1) / (5*(n+1)):
%p A377665 seq(a(n), n = 0..18);
%t A377665 a[n_] := 5^n (4^(n+1) HurwitzZeta[-n, 1/(20)] - 2^(n + 1) HurwitzZeta[-n, 1/(10)]);
%t A377665 Table[Round[N[a[n], 64]], {n, 0, 18}]
%o A377665 (SageMath)
%o A377665 from mpmath import *
%o A377665 mp.dps = 32; mp.pretty = True
%o A377665 def a(n): return int(imag(2*I*(1+sum(binomial(n, j)*polylog(-j, I)*5^j for j in range(n+1)))))
%o A377665 print([a(n) for n in range(19)])
%Y A377665 Cf. A377666.
%K A377665 sign
%O A377665 0,2
%A A377665 _Peter Luschny_, Nov 13 2024