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
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
A353913
Product_{n>=1} 1 / (1 - a(n)*x^n/n!) = 1 + arcsin(x).
Original entry on oeis.org
1, -2, 1, -28, 29, -194, 1583, -61328, 144153, -1697262, 20127867, -191762088, 3978820221, -66586416948, 1057400360235, -58260102945024, 370244721585681, -7992573879248406, 162968423791332339, -3399970067764816824, 88052648301403014789, -2360852841450177138924
Offset: 1
-
nn = 22; f[x_] := Product[1/(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
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
A353972
Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + arcsin(x).
Original entry on oeis.org
1, 0, 1, -4, 29, -124, 1583, -17088, 124553, -1152816, 20127867, -262838016, 3978820221, -48595514304, 914656587063, -24441484099584, 370244721585681, -5884988565575424, 162968423791332339, -3855257807841017856, 82014901819948738629, -1934570487417807744000, 58311771938510122952559
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 - 2)!!/(n (n - 1)!!) - b[n, n - 1]; a[n_] := n! c[n]; Table[a[n], {n, 1, 23}]
Showing 1-6 of 6 results.