A354016 Product_{n>=1} (1 + x^n/n!)^a(n) = exp(x).

1, 1, -2, 9, -24, 70, -720, 5985, -39200, 337176, -3628800, 40907790, -479001600, 6128488080, -87104969952, 1318070979225, -20922789888000, 354250929192160, -6402373705728000, 121882099274319384, -2432849766865689600, 51041047393559059200

Offset

1,3

Links

Table of n, a(n) for n=1..22.

Formula

a(1) = 1; a(n) = (n-1)! * ((-1)^n + Sum_{d|n, 1 < d < n} d * (-d!)^(-n/d) * a(d)).

Mathematica

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}]

Crossrefs

Cf. A006973, A137852, A170910, A170911, A353822.

Sequence in context: A351252 A275260 A248436 * A222667 A122006 A200086

Adjacent sequences: A354013 A354014 A354015 * A354017 A354018 A354019

Keyword

sign

Author

Ilya Gutkovskiy, May 14 2022

Status

approved