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}]
A137941
G.f.: exp(x) = Product_{n>=1} [1 + a(2n-1)*x^(2n-1)/(2n-1)! + a(2n)*x^(2n)/(2n)! ].
Original entry on oeis.org
1, 1, 1, -3, 6, 10, -195, 945, -3080, 17136, 37170, -796950, 3880800, 122007600, -2380685307, 18531377865, -136619683200, 942469528000, 723819456360, -118481659031736, 1884943573381440, -16067035945724400, -323306803861791750, 15940746503392961250
Offset: 1
exp(x) = (1 + 1*x + 1*x^2/2!) * (1 + 1*x^3/3! - 3*x^4/4!) * (1 + 6*x^5/5! + 10*x^6/6!) * (1 - 195*x^7/7! + 945*x^8/8!) *...
-
{a(n)=if(n<1, 0, n!*polcoeff(exp(x +x*O(x^n))/prod(k=1, (n-1)\2, 1+a(2*k-1)*x^(2*k-1)/(2*k-1)!+a(2*k)*x^(2*k)/(2*k)! +x*O(x^n)), n))}
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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