A348205
Product_{n>=1} 1 / (1 - a(n)*x^n/n!) = 1 + log(1 + x).
Original entry on oeis.org
1, -3, 5, -68, 204, -1394, 16862, -413776, 2377512, -35594832, 558727872, -8067263280, 185546362416, -4108304962176, 82441247589360, -3519099528152064, 50908186083448320, -1465023121035418368, 38998680958184088960, -1219845314470474404864, 36452994894649858339584
Offset: 1
A348206
Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + log(1 + x).
Original entry on oeis.org
1, -1, 5, -26, 204, -1434, 16862, -166536, 2377512, -29870400, 558727872, -8542202976, 185546362416, -3332732184768, 82441247589360, -1824937537167744, 50908186083448320, -1214743725939310848, 38998680958184088960, -1084067907183602910720, 36452994894649858339584
Offset: 1
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
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
A353820
Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + arctan(x).
Original entry on oeis.org
1, 0, -2, 8, -16, 96, -832, 9344, -27648, 238080, -4228608, 55812096, -398991360, 4930609152, -98606039040, 2440552022016, -17762113880064, 235149341884416, -7331825098948608, 170578782435409920, -2009778629489197056, 38563016760590598144, -1278044473427380666368
Offset: 1
-
nn = 23; f[x_] := Product[(1 + a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - ArcTan[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
A353821
Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + arctanh(x).
Original entry on oeis.org
1, 0, 2, -8, 64, -384, 3968, -34432, 414720, -4454400, 68247552, -912236544, 15949529088, -245572583424, 5012834549760, -92436465352704, 2119956936523776, -42836227522560000, 1123874181449515008, -26161653829651660800, 730049769522063212544, -18719979459270521389056
Offset: 1
-
nn = 22; f[x_] := Product[(1 + a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - ArcTanh[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
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
A006177
Witt vector *2!/2!.
Original entry on oeis.org
1, 1, 3, 8, 25, 72, 245, 772, 2692, 8925, 32065, 109890, 400023, 1402723, 5165327, 18484746, 68635477, 248339122, 930138521, 3406231198, 12810761323, 47306348881, 178987624513, 665627041157, 2528210175630, 9456885664122
Offset: 1
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- H. Gaudier, Relèvement des coefficients binomiaux dans les vecteurs de Witt, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 16 (1988), 358/S-18, pp. 93-108.
- H. Gaudier, Relèvement des coefficients binomiaux dans les vecteurs de Witt, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 16 (1988), 358/S-18, pp. 93-107. (Annotated scanned copy)
A353822
Product_{n>=1} (1 + x^n/n!)^a(n) = exp(-x)/(1 - x).
Original entry on oeis.org
0, 1, 2, 9, 24, 110, 720, 5985, 39200, 343224, 3628800, 41295870, 479001600, 6130959120, 87104969952, 1318070979225, 20922789888000, 354344089779680, 6402373705728000, 121882240625961816, 2432849766865689600, 51041049953430700800, 1124000727777607680000
Offset: 1
-
nn = 23; f[x_] := Product[(1 + x^n/n!)^a[n], {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - Exp[-x]/(1 - x), {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
Comments