A354277
Product_{n>=1} 1 / (1 - x^n/n!)^a(n) = exp(-x) / (1 - x).
Original entry on oeis.org
0, 1, 2, 3, 24, 70, 720, 4305, 39200, 337176, 3628800, 38417610, 479001600, 6128488080, 87104969952, 1297383162075, 20922789888000, 354250929192160, 6402373705728000, 121407227453840328, 2432849766865689600, 51041047393559059200, 1124000727777607680000
Offset: 1
-
a[1] = 0; a[n_] := a[n] = (n - 1)! (1 - Sum[d d!^(-n/d) a[d], {d, Divisors[n]~Complement~{1, n}}]); Table[a[n], {n, 1, 23}]
A354278
Product_{n>=1} 1 / (1 - a(n)*x^n/n!) = exp(-x) / (1 - x).
Original entry on oeis.org
0, 1, 2, 3, 24, 50, 720, 4095, 35840, 267624, 3628800, 35724150, 479001600, 5240149200, 82614884352, 1188272460375, 20922789888000, 320893244672000, 6402373705728000, 113803149223980216, 2379913632645120000, 46396417566975840000, 1124000727777607680000
Offset: 1
-
a[1] = 0; a[n_] := a[n] = (n - 1)! (1 - Sum[d (a[d]/d!)^(n/d), {d, Divisors[n]~Complement~{1, n}}]); Table[a[n], {n, 1, 23}]
A354016
Product_{n>=1} (1 + x^n/n!)^a(n) = exp(x).
Original entry on oeis.org
1, 1, -2, 9, -24, 70, -720, 5985, -39200, 337176, -3628800, 40907790, -479001600, 6128488080, -87104969952, 1318070979225, -20922789888000, 354250929192160, -6402373705728000, 121882099274319384, -2432849766865689600, 51041047393559059200
Offset: 1
-
nn = 22; f[x_] := Product[(1 + x^n/n!)^a[n], {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - Exp[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
a[1] = 1; a[n_] := a[n] = (n - 1)! ((-1)^n + Sum[d (-d!)^(-n/d) a[d], {d, Divisors[n] ~ Complement ~ {1, n}}]); Table[a[n], {n, 1, 22}]
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