A355381 Expansion of e.g.f. exp(exp(3*x) - exp(2*x)).

1, 1, 6, 35, 247, 2102, 20547, 224541, 2707292, 35638329, 507464939, 7757439428, 126538995293, 2191454313661, 40120212534838, 773554002955047, 15656660861190371, 331700076893737054, 7337160433117899959, 169068422994937678185, 4050093664805130165348

Offset: 0

Author

Vaclav Kotesovec, Jun 30 2022

Keywords

• nonn

Comments

In general, if m > 0, b > d >= 1 and e.g.f. = exp(m*exp(b*x) + r*exp(d*x) + s) then a(n) ~ exp(m*exp(b*z) + r*exp(d*z) + s - n) * (n/z)^(n + 1/2) / sqrt(m*b*(1 + b*z)*exp(b*z) + r*d*(1 + d*z)*exp(d*z)), where z = LambertW(n/m)/b - 1/(d + b/LambertW(n/m) + b^2 * m^(d/b) * n^(1 - d/b) * (1 + LambertW(n/m)) / (d*r*LambertW(n/m)^(2 - d/b))). - Vaclav Kotesovec, Jul 03 2022

In addition, if b/d >=2 then a(n) ~ c * (b*n/LambertW(n/m))^n * exp(n/LambertW(n/m) + r * (n/(m*LambertW(n/m)))^(d/b) - n + s) / sqrt(1 + LambertW(n/m)), where c = 1 for b/d > 2 and c = exp(-r^2/(8*m)) for b/d = 2. - Vaclav Kotesovec, Jul 10 2022

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..471

Crossrefs

Cf. A143405, A194006, A355291, A355380, A355378.

Programs

Mathematica

nmax = 20; CoefficientList[Series[Exp[Exp[3*x] - Exp[2*x]], {x, 0, nmax}], x] * Range[0, nmax]!
Table[Sum[Binomial[n,k] * 3^k * 2^(n-k) * BellB[k] * BellB[n-k, -1], {k, 0, n}], {n, 0, 20}]

PARI

my(x='x+O('x^25)); Vec(serlaplace(exp(exp(3*x) - exp(2*x)))) \\ Michel Marcus, Jun 30 2022

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * 3^k * 2^(n-k) * Bell(k) * Bell(n-k, -1).

a(0) = 1; a(n) = Sum_{k=1..n} (3^k - 2^k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Jun 30 2022

a(n) ~ exp(exp(3*z) - exp(2*z) - n) * (n/z)^(n + 1/2) / sqrt(3*(1 + 3*z)*exp(3*z) - 2*(1 + 2*z)*exp(2*z)), where z = LambertW(n)/3 - 1/(2 + 3/LambertW(n) - 9 * n^(1/3) * (1 + LambertW(n)) / (2*LambertW(n)^(4/3))). - Vaclav Kotesovec, Jul 03 2022