A352691
Product_{n>=1} 1 / (1 - x^n)^(a(n)/n!) = 1 + log(1 + x).
Original entry on oeis.org
1, -3, 5, -23, 204, -1894, 16862, -166466, 2346712, -37858296, 558727872, -9031080288, 185546362416, -3960341036352, 83728926109488, -1961110591316304, 50908186083448320, -1384998141007364736, 38998680958184088960, -1160052698286814237056, 37029733866954589964544
Offset: 1
-
nmax = 21; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + Log[1 + x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
A352664
Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 - log(1 - x).
Original entry on oeis.org
1, 1, -1, 13, -16, -34, -526, 22142, -10424, -160536, -2805408, -29182944, -374664720, -3220913760, 32949033168, 11465880121776, -16610113920768, -96543735968640, -5110200130727808, -130871898552663936, 1042965176555775744, -29461082210774712576
Offset: 1
-
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[c[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; c[n_] := c[n] = 1/n - b[n, n - 1]; a[n_] := n! c[n]; Table[a[n], {n, 1, 22}]
A352953
Product_{n>=1} 1 / (1 - x^n)^(a(n)/n!) = 1 - log(1 - x).
Original entry on oeis.org
1, -1, -1, 1, -16, 86, -526, 302, -10424, 323304, -2805408, -6563424, -374664720, 5877455520, 32949033168, -24011091024, -16610113920768, 87369247685760, -5110200130727808, -23241729685643136, 1042965176555775744, 49535245994720788224
Offset: 1
-
nmax = 22; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 - Log[1 - x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
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