A353818
Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + arcsin(x).
Original entry on oeis.org
1, 0, 1, -4, 29, -174, 1583, -13168, 144153, -1485330, 20127867, -253341144, 3978820221, -57986205900, 1057400360235, -18016221644544, 370244721585681, -6993826454599146, 162968423791332339, -3490951922268853320, 88052648301403014789, -2075060448716599488276
Offset: 1
-
nn = 22; f[x_] := Product[(1 + a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - ArcSin[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
A353819
Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + arcsinh(x).
Original entry on oeis.org
1, 0, -1, 4, -11, 66, -547, 4880, -27351, 263310, -3258663, 39791016, -390445563, 5477278548, -84140635815, 1486404086016, -18431412645519, 322018685539542, -6436900596281679, 133183534639917240, -2208721087854287811, 49383164607876494604, -1149793471388581053219
Offset: 1
-
nn = 23; f[x_] := Product[(1 + a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - ArcSinh[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
A353820
Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + arctan(x).
Original entry on oeis.org
1, 0, -2, 8, -16, 96, -832, 9344, -27648, 238080, -4228608, 55812096, -398991360, 4930609152, -98606039040, 2440552022016, -17762113880064, 235149341884416, -7331825098948608, 170578782435409920, -2009778629489197056, 38563016760590598144, -1278044473427380666368
Offset: 1
-
nn = 23; f[x_] := Product[(1 + a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - ArcTan[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
A354118
Product_{n>=1} 1 / (1 - x^n)^(a(n)/n!) = 1 + arctanh(x).
Original entry on oeis.org
1, -2, 2, -8, 64, -544, 3968, -29952, 378880, -5938176, 68247552, -793491456, 15949529088, -306908848128, 4760383438848, -90615249567744, 2119956936523776, -49428158281678848, 1123874181449515008, -26217392043061149696, 722523072906903158784, -21323712124731229470720, 589068777481530305937408
Offset: 1
-
nmax = 23; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + ArcTanh[x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
A354276
Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + arctanh(x).
Original entry on oeis.org
1, 0, 2, -8, 64, -304, 3968, -43392, 378880, -4002816, 68247552, -995736576, 15949529088, -238273241088, 4760383438848, -113132156780544, 2119956936523776, -42743492966350848, 1123874181449515008, -28901050300546154496, 722523072906903158784, -19401957422023594475520, 589068777481530305937408
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] = Mod[n, 2]/n - b[n, n - 1]; a[n_] := n! c[n]; Table[a[n], {n, 1, 23}]
A353928
Product_{n>=1} 1 / (1 - a(n)*x^n/n!) = 1 + arctanh(x).
Original entry on oeis.org
1, -2, 2, -32, 64, -464, 3968, -92672, 414720, -5486592, 68247552, -869895168, 15949529088, -299609505792, 5012834549760, -177156842717184, 2119956936523776, -50954009373573120, 1123874181449515008, -29311973327486582784, 730049769522063212544, -22005690087484302557184
Offset: 1
-
nn = 22; f[x_] := Product[1/(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
Showing 1-6 of 6 results.