A354066
Product_{n>=1} 1 / (1 - x^n)^(a(n)/n!) = 1 + tanh(x).
Original entry on oeis.org
1, -2, -2, 8, -24, 224, -720, -1408, 0, 717824, -3628800, -47546368, -479001600, 12431673344, 87178291200, -68669145088, -20922789888000, 47215125069824, -6402373705728000, -159504062197792768, 2432902008176640000, 102176932845365755904, -1124000727777607680000
Offset: 1
-
nmax = 23; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + Tanh[x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
A353821
Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + arctanh(x).
Original entry on oeis.org
1, 0, 2, -8, 64, -384, 3968, -34432, 414720, -4454400, 68247552, -912236544, 15949529088, -245572583424, 5012834549760, -92436465352704, 2119956936523776, -42836227522560000, 1123874181449515008, -26161653829651660800, 730049769522063212544, -18719979459270521389056
Offset: 1
-
nn = 22; f[x_] := Product[(1 + a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - ArcTanh[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
A353912
Product_{n>=1} 1 / (1 - a(n)*x^n/n!) = 1 + tanh(x).
Original entry on oeis.org
1, -2, -2, -16, -24, 64, -720, -23808, -35840, 282368, -3628800, -75458560, -479001600, 5315078144, -82614884352, -8601835798528, -20922789888000, 321288633450496, -6402373705728000, -309168395474436096, -2379913632645120000, 46441359567137275904
Offset: 1
-
nn = 22; f[x_] := Product[1/(1 - a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - Tanh[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
A354176
Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + tanh(x).
Original entry on oeis.org
1, 0, -2, 8, -24, -16, -720, 12032, 0, -7936, -3628800, -58190848, -479001600, -22368256, 87178291200, 6174957043712, -20922789888000, 47215125069824, -6402373705728000, -164824694455533568, 2432902008176640000, -4951498053124096, -1124000727777607680000
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] = 2^(n + 1) (2^(n + 1) - 1) BernoulliB[n + 1]/((n + 1) n!) - b[n, n - 1]; a[n_] := n! c[n]; Table[a[n], {n, 1, 23}]
Showing 1-4 of 4 results.