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}]
A354175
Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + tan(x).
Original entry on oeis.org
1, 0, 2, -8, 56, -256, 3184, -36224, 273920, -2845696, 48104704, -676312064, 10591523840, -149454094336, 2888557717504, -72214957359104, 1249919350046720, -23620669488234496, 624022403933077504, -15637185047733469184, 372737701735949926400, -9655667879651150135296
Offset: 1
Cf.
A000182,
A009006,
A067856,
A353583,
A353584,
A353611,
A353911,
A354065,
A354171,
A354172,
A354173,
A354174,
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] = 2^(n + 1) (2^(n + 1) - 1) Abs[BernoulliB[n + 1]]/((n + 1) n!) - b[n, n - 1]; a[n_] := n! c[n]; Table[a[n], {n, 1, 22}]
A354176
Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + tanh(x).
Original entry on oeis.org
1, 0, -2, 8, -24, -16, -720, 12032, 0, -7936, -3628800, -58190848, -479001600, -22368256, 87178291200, 6174957043712, -20922789888000, 47215125069824, -6402373705728000, -164824694455533568, 2432902008176640000, -4951498053124096, -1124000727777607680000
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] = 2^(n + 1) (2^(n + 1) - 1) BernoulliB[n + 1]/((n + 1) n!) - b[n, n - 1]; a[n_] := n! c[n]; Table[a[n], {n, 1, 23}]
A354173
Product_{n>=1} (1 + x^(2*n))^(a(n)/(2*n)!) = cos(x).
Original entry on oeis.org
-1, -8, 104, -12032, 354944, 47546368, 6204652544, -6174957043712, 47215125069824, 159504062197792768, 51085990673656315904, 54592541528151763714048, 15510963121850795776016384, 14479308135716773591282352128, -7469518701197092988127633473536, -77646018400552596699424746364731392
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] = {1, 0, -1, 0}[[Mod[n + 1, 4, 1]]]/n! - b[n, n - 1]; a[n_] := (2 n)! c[2 n]; Table[a[n], {n, 1, 16}]
A354174
Product_{n>=1} (1 + x^(2*n))^(a(n)/(2*n)!) = cosh(x).
Original entry on oeis.org
1, 4, -104, 8128, -354944, -21642752, -6204652544, 4286437900288, -47215125069824, -78465506362130432, -51085990673656315904, -35027783166649488637952, -15510963121850795776016384, -7220202338641080038690127872, 7469518701197092988127633473536, 53919400066294168384184259715268608
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 + 1, 2]/n! - b[n, n - 1]; a[n_] := (2 n)! c[2 n]; Table[a[n], {n, 1, 16}]
A354274
Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + arcsinh(x).
Original entry on oeis.org
1, 0, -1, 4, -11, -4, -547, 7680, -7751, 81744, -3258663, -9474816, -390445563, 233029824, -964154427, 4193551958016, -18431412645519, 71090090006784, -6436900596281679, 17349989459410944, 834261829219880829, -241960391975347200, -1149793471388581053219
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] = {1, 0, -1, 0}[[Mod[n, 4, 1]]] (n - 2)!!/(n (n - 1)!!) - b[n, n - 1]; a[n_] := n! c[n]; Table[a[n], {n, 1, 23}]
Showing 1-6 of 6 results.