A354056
Product_{n>=1} 1 / (1 - x^n)^(a(n)/n!) = 1 + sinh(x).
Original entry on oeis.org
1, -2, 1, -4, 21, -196, 1023, -5440, 65145, -1237456, 10925883, -69882880, 1994183205, -39099282496, 372390766023, -6270496768000, 158096182329585, -3268815510804736, 64115697136312563, -1009052458754375680, 27389518837925527965, -924645800211698308096, 19391677044464348893503
Offset: 1
-
nmax = 23; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + Sinh[x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
A354065
Product_{n>=1} 1 / (1 - x^n)^(a(n)/n!) = 1 + tan(x).
Original entry on oeis.org
1, -2, 2, -8, 56, -496, 3184, -22784, 273920, -4539136, 48104704, -506000384, 10591523840, -204528633856, 2888557717504, -53417657237504, 1249919350046720, -28453501844586496, 624022403933077504, -13729309300086800384, 372737701735949926400, -11010228423219933085696
Offset: 1
Cf.
A000182,
A003707,
A009006,
A353583,
A353584,
A353611,
A353911,
A354055,
A354056,
A354063,
A354064,
A354066.
-
nmax = 22; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + Tan[x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
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
A354063
Product_{n>=1} 1 / (1 - x^(2*n))^(a(n)/(2*n)!) = cos(x).
Original entry on oeis.org
-1, 4, 104, 1408, 354944, -21642752, 6204652544, 68669145088, 47215125069824, -78465506362130432, 51085990673656315904, -6994033618612756938752, 15510963121850795776016384, -7220202338641080038690127872, -7469518701197092988127633473536, 11962377309169877924807975108608
Offset: 1
-
nmax = 32; Take[CoefficientList[Series[Sum[MoebiusMu[k] Log[Cos[x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}] // Rest
A354064
Product_{n>=1} 1 / (1 - x^(2*n))^(a(n)/(2*n)!) = cosh(x).
Original entry on oeis.org
1, -8, -104, 1408, -354944, 47546368, -6204652544, 68669145088, -47215125069824, 159504062197792768, -51085990673656315904, -6994033618612756938752, -15510963121850795776016384, 14479308135716773591282352128, 7469518701197092988127633473536, 11962377309169877924807975108608
Offset: 1
-
nmax = 32; Take[CoefficientList[Series[Sum[MoebiusMu[k] Log[Cosh[x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}] // Rest
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}]
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
Showing 1-7 of 7 results.