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
A353915
Product_{n>=1} 1 / (1 - a(n)*x^n/n!) = 1 + arctan(x).
Original entry on oeis.org
1, -2, -2, -16, -16, 16, -832, -22016, -27648, 173568, -4228608, -57965568, -398991360, 2554896384, -98606039040, -7568860053504, -17762113880064, 200091412463616, -7331825098948608, -258326401420099584, -2009778629489197056, 25949098553870647296, -1278044473427380666368
Offset: 1
-
nn = 23; f[x_] := Product[1/(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
A354116
Product_{n>=1} 1 / (1 - x^n)^(a(n)/n!) = 1 + arcsinh(x).
Original entry on oeis.org
1, -2, -1, 4, -11, 116, -547, 960, -7751, 414384, -3258663, -6813696, -390445563, 9694641984, -964154427, 208258646016, -18431412645519, 207842731632384, -6436900596281679, -37454668211552256, 834261829219880829, 91517388643567641600, -1149793471388581053219
Offset: 1
-
nmax = 23; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + ArcSinh[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}]
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 1-5 of 5 results.