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%I #83 Jan 15 2021 21:25:21
%S 3,7,23,103,599,4327,37463,378343,4366679,56698087,817980503,
%T 12981060583,224732540759,4214866787047,85130743763543,
%U 1842265527822823,42525237455850839,1042966136233087207,27084277306054762583,742412698554627289063,21421502369955073624919
%N a(n) = 1 + Sum_{m >= 1} (m + 1)^n/2^(m - 1).
%F a(n + 1) = 4*A162509(n + 1) + a(n).
%F a(n) = 2*A007047(n) + 1.
%F {a(4n - 3), a(4n - 2), a(4n - 1), a(4n)} mod 10 = {7, 3, 3, 9} for n > 0.
%F floor(log_2(a(n))) = A083652(n).
%F Lim_{n->infinity} (a(n)^(1/n))/n = 1/(e*log(2)). - _Jon E. Schoenfield_, Feb 24 2018
%F a(n)/n! ~ 4 / (log(2))^(n+1). - _Vaclav Kotesovec_, Apr 17 2018
%t Table[1 + LerchPhi[1/2, -n, 2], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 17 2018 *)
%o (PARI) a(n) = 1+ round(suminf(m=1, (m + 1)^n/2^(m - 1)));
%Y Cf. A007047, A083652, A162509.
%K nonn
%O 0,1
%A _Joseph Wheat_, Feb 20 2018
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