A353250
a(0) = 1, a(n) = harmonic_mean(a(n-1), n), where harmonic_mean(p, q) = 2/(1/p + 1/q); numerators.
Original entry on oeis.org
1, 1, 4, 24, 48, 480, 960, 13440, 26880, 161280, 322560, 7096320, 14192640, 369008640, 738017280, 295206912, 590413824, 20074070016, 40148140032, 1525629321216, 15256293212160, 30512586424320, 61025172848640, 2807157951037440, 5614315902074880
Offset: 0
a(0) = 1,
a(1) = 2/(1/1 + 1/1) = 1,
a(2) = 2/(1/1 + 1/2) = 4/3,
a(3) = 2/(1/(4/3) + 1/3) = 24/13,
a(4) = 2/(1/(24/13) + 1/4) = 48/19, etc.
This sequence gives the numerators: 1, 1, 4, 24, 48, ...
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Table[1/(1/2^n - Re[LerchPhi[2, 1, n + 1]]), {n, 0, 24}] // Numerator (* or *)
a[0] = 1; a[n_Integer] := a[n] = 2/(1/a[n-1] + 1/n); Table[a[n], {n, 0, 24}] // Numerator
A353251
a(0) = 1, a(n) = harmonic_mean(a(n-1), n), where harmonic_mean(p, q) = 2/(1/p + 1/q); denominators.
Original entry on oeis.org
1, 1, 3, 13, 19, 143, 223, 2521, 4201, 21563, 37691, 737161, 1328521, 31463413, 57821173, 21404465, 39854897, 1267947073, 2383173185, 85428430547, 808549483039, 1535039635999, 2921975382559, 128230606647497, 245195521274057, 2348840786785261, 4508193056814061
Offset: 0
a(0) = 1,
a(1) = 2/(1/1 + 1/1) = 1,
a(2) = 2/(1/1 + 1/2) = 4/3,
a(3) = 2/(1/(4/3) + 1/3) = 24/13,
a(4) = 2/(1/(24/13) + 1/4) = 48/19, etc.
This sequence gives the denominators: 1, 1, 3, 13, 19, ...
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Table[1/(1/2^n - Re[LerchPhi[2, 1, n + 1]]), {n, 0, 26}] // Denominator (* or *)
a[0] = 1; a[n_Integer] := a[n] = 2/(1/a[n-1] + 1/n); Table[a[n], {n, 0, 26}] // Denominator
A357591
Expansion of e.g.f. (exp(x) - 1) * tan((exp(x) - 1)/2).
Original entry on oeis.org
0, 0, 1, 3, 8, 25, 99, 476, 2643, 16575, 116002, 895719, 7554311, 69051034, 679913073, 7174562327, 80765185416, 966076987581, 12235992073975, 163590477924708, 2302288709067167, 34021599945907915, 526690307104399482, 8524372522971447683, 143963947160570293851
Offset: 0
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nmax = 20; CoefficientList[Series[(Exp[x] - 1)*Tan[(Exp[x] - 1)/2] , {x, 0, nmax}], x] * Range[0, nmax]!
Table[2*Sum[(-1)^k * StirlingS2[n, 2*k] * (1 - 4^k) * BernoulliB[2*k], {k, 0, n/2}], {n, 0, 20}]
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my(N=30, x='x+O('x^N)); concat([0, 0], Vec(serlaplace((exp(x)-1)*tan((exp(x)-1)/2)))) \\ Seiichi Manyama, Oct 05 2022
A357594
Expansion of e.g.f. log(1-x) * tan(log(1-x)/2).
Original entry on oeis.org
0, 0, 1, 3, 12, 60, 362, 2562, 20820, 191088, 1955020, 22061380, 272197160, 3645227040, 52656804440, 816114251400, 13508168448400, 237805776169600, 4436759277524400, 87445191383773200, 1815460566861236000, 39600109151685600000, 905416958295793788000
Offset: 0
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my(N=30, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(log(1-x)*tan(log(1-x)/2))))
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a(n) = 2*sum(k=0, n\2, (-1)^k*(1-4^k)*abs(stirling(n, 2*k, 1))*bernfrac(2*k));
Showing 1-4 of 4 results.