A354066
Product_{n>=1} 1 / (1 - x^n)^(a(n)/n!) = 1 + tanh(x).
Original entry on oeis.org
1, -2, -2, 8, -24, 224, -720, -1408, 0, 717824, -3628800, -47546368, -479001600, 12431673344, 87178291200, -68669145088, -20922789888000, 47215125069824, -6402373705728000, -159504062197792768, 2432902008176640000, 102176932845365755904, -1124000727777607680000
Offset: 1
-
nmax = 23; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + Tanh[x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
A353873
Product_{n>=1} 1 / (1 - a(n)*x^n/n!) = 1 + sin(x).
Original entry on oeis.org
1, -2, -1, -20, -19, 94, -659, -29392, -38375, 309458, -3578279, -31878824, -476298835, 5459426348, -85215100151, -12006576849152, -20903398375855, 314905758207466, -6399968826052559, -178647405711887800, -2394435177245209195, 46569786580097365748
Offset: 1
-
nn = 22; f[x_] := Product[1/(1 - a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - Sin[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
A353910
Product_{n>=1} 1 / (1 - a(n)*x^n/n!) = 1 + sinh(x).
Original entry on oeis.org
1, -2, 1, -28, 21, -146, 1023, -56400, 84745, -975502, 10925883, -57795112, 1994183205, -32047567540, 489891177051, -43944425632000, 158096182329585, -3254060029210454, 64115697136312563, -921897484040044728, 31920276313015362525, -812922524976721463020
Offset: 1
-
nn = 22; f[x_] := Product[1/(1 - a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - Sinh[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
A353911
Product_{n>=1} 1 / (1 - a(n)*x^n/n!) = 1 + tan(x).
Original entry on oeis.org
1, -2, 2, -32, 56, -416, 3184, -85504, 309760, -4087552, 48104704, -546922496, 10591523840, -194387924992, 3133776259072, -129880886411264, 1249919350046720, -29073986250604544, 624022403933077504, -15137719350365519872, 381632216575339397120, -11149155036737662615552
Offset: 1
-
nn = 22; f[x_] := Product[1/(1 - a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - Tan[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
A354176
Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + tanh(x).
Original entry on oeis.org
1, 0, -2, 8, -24, -16, -720, 12032, 0, -7936, -3628800, -58190848, -479001600, -22368256, 87178291200, 6174957043712, -20922789888000, 47215125069824, -6402373705728000, -164824694455533568, 2432902008176640000, -4951498053124096, -1124000727777607680000
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] = 2^(n + 1) (2^(n + 1) - 1) BernoulliB[n + 1]/((n + 1) 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.