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}]
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}]
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}]
A354275
Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + arctan(x).
Original entry on oeis.org
1, 0, -2, 8, -16, -64, -832, 13824, 8192, -36096, -4228608, -58438656, -398991360, -3452915712, 44581613568, 7144463302656, -17762113880064, 126440605483008, -7331825098948608, -88237584523984896, 3154526750647517184, -27279757707305287680, -1278044473427380666368
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 - b[n, n - 1]; a[n_] := n! c[n]; Table[a[n], {n, 1, 23}]
Showing 1-6 of 6 results.