A355337 Expansion of e.g.f.: exp(exp(x) + x^2 - 1).

1, 1, 4, 11, 51, 212, 1133, 6001, 36508, 228435, 1559575, 11079180, 83753497, 659858617, 5459331036, 46980355355, 421272977267, 3917446787884, 37766791690501, 376447420971545, 3875957531387172, 41149332371734371, 449984429580538407, 5061923434006018612, 58517321729774406129

Offset: 0

Author

Vaclav Kotesovec, Jun 29 2022

Keywords

• nonn

Comments

In general, if m >= 1, b <> 0 and e.g.f. = exp(m*exp(b*x) + q*x^2 + r*x + s) then a(n) ~ b^n * n^(n + r/b) * exp(n/LambertW(n/m) + q*LambertW(n/m)^2 / b^2 - n + s) / (m^(r/b) * sqrt(1 + LambertW(n/m)) * LambertW(n/m)^(n + r/b)).

Number of ways the roots of a polynomial with real coefficients and degree n can be configured regarding multiplicity and complexity. By configuration we mean for example a product of the form (x-b)*(x-c)*...; the roots of a polynomial do not imply any order, but the parameters which define roots may be labeled. In the case of a conjugate complex pair, we will distinguish between positive and negative imaginary part. For details see the example for a(4) in the "LINKS" section. - Thomas Scheuerle, Jun 01 2024

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..568
• Thomas Scheuerle, An example for a(4) = 51 which explains the relation to polynomial roots.

Crossrefs

Cf. A000110, A277381, A316778, A355338.

Programs

Mathematica

nmax = 25; CoefficientList[Series[Exp[Exp[x] + x^2 - 1], {x, 0, nmax}], x] * Range[0, nmax]!

PARI

my(x='x+O('x^30)); Vec(serlaplace(exp(exp(x) + x^2 - 1))) \\ Michel Marcus, Jun 29 2022

PARI

a(n) = sum(k=0, floor(n/2), sum(m=0, n-(k*2), stirling(n-(k*2), m, 2))*(2*k)!/k!*binomial(n, n-(k*2))) \\ Thomas Scheuerle, Jun 01 2024

Formula

a(n) ~ n^n * exp(n/LambertW(n) + LambertW(n)^2 - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^n).

a(n) ~ Bell(n) * exp(LambertW(n)^2).

a(0) = a(1) = 1; a(n) = 2 * (n-1) * a(n-2) + Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Jun 29 2022

a(n) = Sum_{k=0..floor(n/2)} A000110(n - k*2)*(2*k)!/k!*binomial(n, n - (k*2)). - Thomas Scheuerle, Jun 01 2024