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A353250 a(0) = 1, a(n) = harmonic_mean(a(n-1), n), where harmonic_mean(p, q) = 2/(1/p + 1/q); numerators.

%I A353250 #14 Apr 09 2022 02:25:00
%S A353250 1,1,4,24,48,480,960,13440,26880,161280,322560,7096320,14192640,
%T A353250 369008640,738017280,295206912,590413824,20074070016,40148140032,
%U A353250 1525629321216,15256293212160,30512586424320,61025172848640,2807157951037440,5614315902074880
%N A353250 a(0) = 1, a(n) = harmonic_mean(a(n-1), n), where harmonic_mean(p, q) = 2/(1/p + 1/q); numerators.
%H A353250 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicMean.html">Harmonic Mean</a>.
%H A353250 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LerchTranscendent.html">Lerch Transcendent</a>.
%H A353250 Wikipedia, <a href="https://en.wikipedia.org/wiki/Harmonic_mean">Harmonic mean</a>.
%F A353250 a(n) = numerator(1/(1/2^n - Re(Phi(2, 1, n+1)))), where Phi(z, s, a) is the Lerch transcendent.
%e A353250 a(0) = 1,
%e A353250 a(1) = 2/(1/1 + 1/1) = 1,
%e A353250 a(2) = 2/(1/1 + 1/2) = 4/3,
%e A353250 a(3) = 2/(1/(4/3) + 1/3) = 24/13,
%e A353250 a(4) = 2/(1/(24/13) + 1/4) = 48/19, etc.
%e A353250 This sequence gives the numerators: 1, 1, 4, 24, 48, ...
%t A353250 Table[1/(1/2^n - Re[LerchPhi[2, 1, n + 1]]), {n, 0, 24}] // Numerator (* or *)
%t A353250 a[0] = 1; a[n_Integer] := a[n] = 2/(1/a[n-1] + 1/n); Table[a[n], {n, 0, 24}] // Numerator
%Y A353250 Cf. A353251 (denominators).
%Y A353250 Cf. A003149, A136128, A191778 (has many terms in common), A241519, A242376.
%K A353250 nonn,frac
%O A353250 0,3
%A A353250 _Vladimir Reshetnikov_, Apr 08 2022