A353608
Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + sinh(x).
Original entry on oeis.org
1, 0, 1, -4, 21, -126, 1023, -8240, 84745, -864370, 10925883, -133566808, 1994183205, -28455880012, 489891177051, -8112780640000, 158096182329585, -2911196026492074, 64115697136312563, -1328879415116924744, 31920276313015362525, -728711636884140292372
Offset: 1
-
nn = 22; f[x_] := Product[(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
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
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
A353818
Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + arcsin(x).
Original entry on oeis.org
1, 0, 1, -4, 29, -174, 1583, -13168, 144153, -1485330, 20127867, -253341144, 3978820221, -57986205900, 1057400360235, -18016221644544, 370244721585681, -6993826454599146, 162968423791332339, -3490951922268853320, 88052648301403014789, -2075060448716599488276
Offset: 1
-
nn = 22; f[x_] := Product[(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
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
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}]
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.