A353607
Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + sin(x).
Original entry on oeis.org
1, 0, -1, 4, -19, 114, -659, 5328, -38375, 400430, -3578279, 44920360, -476298835, 6949878740, -85215100151, 1492480745728, -20903398375855, 382829285287446, -6399968826052559, 136747967762351544, -2394435177245209195, 55602194767215266060, -1123931378903214542099
Offset: 1
-
nn = 23; f[x_] := Product[(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
A353611
Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + tan(x).
Original entry on oeis.org
1, 0, 2, -8, 56, -336, 3184, -27264, 309760, -3297280, 48104704, -624745472, 10591523840, -159594803200, 3133776259072, -56224864108544, 1249919350046720, -24600643845095424, 624022403933077504, -14094091678163140608, 381632216575339397120, -9516741266133420605440
Offset: 1
-
nn = 22; f[x_] := Product[(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
A353609
Product_{n>=1} (1 + a(n)*x^(2*n)/(2*n)!) = cosh(x).
Original entry on oeis.org
1, 1, -14, 393, -14744, 972610, -74928944, 9322093753, -1163849271296, 228519734620776, -44942000161435904, 12717856972091286642, -3539995034294896016384, 1371560847857743301790928, -510461123036204706738612224, 268938575250382935485761673113
Offset: 1
-
nn = 16; f[x_] := Product[(1 + a[n] x^(2 n)/(2 n)!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - Cosh[x], {x, 0, 2 nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
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
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
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
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}]
A353610
Product_{n>=1} (1 + a(n)*x^(2*n)/(2*n)!) = sec(x).
Original entry on oeis.org
1, 5, -14, 1777, -14744, 247994, -74928944, 42293543177, -1163849271296, 95795966018440, -44942000161435904, 4494117864138588514, -3539995034294896016384, 770158600620174924566672, -510461123036204706738612224, 1162153458061287151457003978297
Offset: 1
-
nn = 16; f[x_] := Product[(1 + a[n] x^(2 n)/(2 n)!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - Sec[x], {x, 0, 2 nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
A353779
Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + tanh(x).
Original entry on oeis.org
1, 0, -2, 8, -24, 144, -720, 7552, -35840, 427520, -3628800, 45415424, -479001600, 7094226944, -82614884352, 1741160087552, -20922789888000, 371094631612416, -6402373705728000, 137529198176370688, -2379913632645120000, 55730621780175355904
Offset: 1
-
nn = 22; f[x_] := Product[(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
Showing 1-9 of 9 results.