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
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
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
A354172
Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + sinh(x).
Original entry on oeis.org
1, 0, 1, -4, 21, -76, 1023, -12160, 65145, -602416, 10925883, -120444160, 1994183205, -21404165056, 372390766023, -12580544512000, 158096182329585, -2119447579092736, 64115697136312563, -1412937791690260480, 27389518837925527965, -616988361649163447296, 19391677044464348893503
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] = Mod[n, 2]/n! - b[n, n - 1]; a[n_] := n! c[n]; Table[a[n], {n, 1, 23}]
A353914
Product_{n>=1} 1 / (1 - a(n)*x^n/n!) = 1 + arcsinh(x).
Original entry on oeis.org
1, -2, -1, -20, -11, 46, -547, -29840, -27351, 232818, -3258663, -29911848, -390445563, 4450393260, -84140635815, -12153983817984, -18431412645519, 286688710444842, -6436900596281679, -169286474970429624, -2208721087854287811, 41892263643715799796, -1149793471388581053219
Offset: 1
-
nn = 23; f[x_] := Product[1/(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
Showing 1-6 of 6 results.