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
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
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
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
A354275
Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + arctan(x).
Original entry on oeis.org
1, 0, -2, 8, -16, -64, -832, 13824, 8192, -36096, -4228608, -58438656, -398991360, -3452915712, 44581613568, 7144463302656, -17762113880064, 126440605483008, -7331825098948608, -88237584523984896, 3154526750647517184, -27279757707305287680, -1278044473427380666368
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 - b[n, n - 1]; a[n_] := n! c[n]; Table[a[n], {n, 1, 23}]
Showing 1-6 of 6 results.