A354055
Product_{n>=1} 1 / (1 - x^n)^(a(n)/n!) = 1 + sin(x).
Original entry on oeis.org
1, -2, -1, 4, -19, 164, -659, 1408, -18775, 642224, -3578279, -21642752, -476298835, 11904106304, 25626362581, 68669145088, -20903398375855, 212840905389824, -6399968826052559, -78465506362130432, 1010700510694925525, 101465632831736751104, -1123931378903214542099
Offset: 1
Cf.
A170914,
A170915,
A328186,
A328191,
A353607,
A353873,
A354056,
A354063,
A354064,
A354065,
A354066.
-
nmax = 23; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + Sin[x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
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
A353912
Product_{n>=1} 1 / (1 - a(n)*x^n/n!) = 1 + tanh(x).
Original entry on oeis.org
1, -2, -2, -16, -24, 64, -720, -23808, -35840, 282368, -3628800, -75458560, -479001600, 5315078144, -82614884352, -8601835798528, -20922789888000, 321288633450496, -6402373705728000, -309168395474436096, -2379913632645120000, 46441359567137275904
Offset: 1
-
nn = 22; f[x_] := Product[1/(1 - a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - Tanh[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
A354171
Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + sin(x).
Original entry on oeis.org
1, 0, -1, 4, -19, 44, -659, 8128, -18775, 67664, -3578279, 7629568, -476298835, 505198784, 25626362581, 4286437900288, -20903398375855, -118410655250176, -6399968826052559, -33100680116191232, 1010700510694925525, 706348515575880704, -1123931378903214542099
Offset: 1
Cf.
A067856,
A170914,
A170915,
A328186,
A328191,
A353607,
A353873,
A354055,
A354172,
A354173,
A354174,
A354175,
A354176.
-
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}]
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
Showing 1-6 of 6 results.