A354115
Product_{n>=1} 1 / (1 - x^n)^(a(n)/n!) = 1 + arcsin(x).
Original entry on oeis.org
1, -2, 1, -4, 29, -244, 1583, -10368, 124553, -2029776, 20127867, -180343296, 3978820221, -75977108544, 914656587063, -15574206480384, 370244721585681, -8082505243732224, 162968423791332339, -3082360882836013056, 82014901819948738629, -2501342802748968883200, 58311771938510122952559
Offset: 1
-
nmax = 23; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + ArcSin[x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
A354117
Product_{n>=1} 1 / (1 - x^n)^(a(n)/n!) = 1 + arctan(x).
Original entry on oeis.org
1, -2, -2, 8, -16, 176, -832, 384, 8192, 447744, -4228608, -15860736, -398991360, 10938421248, 44581613568, -29064658944, -17762113880064, -18092698632192, -7331825098948608, -64037289416196096, 3154526750647517184, 91791873021766533120, -1278044473427380666368
Offset: 1
-
nmax = 23; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + ArcTan[x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
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
A354274
Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + arcsinh(x).
Original entry on oeis.org
1, 0, -1, 4, -11, -4, -547, 7680, -7751, 81744, -3258663, -9474816, -390445563, 233029824, -964154427, 4193551958016, -18431412645519, 71090090006784, -6436900596281679, 17349989459410944, 834261829219880829, -241960391975347200, -1149793471388581053219
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, 0, -1, 0}[[Mod[n, 4, 1]]] (n - 2)!!/(n (n - 1)!!) - b[n, n - 1]; a[n_] := n! c[n]; Table[a[n], {n, 1, 23}]
Showing 1-4 of 4 results.