%I #32 Jun 17 2024 15:51:33

%S 1,1,4,11,51,212,1133,6001,36508,228435,1559575,11079180,83753497,

%T 659858617,5459331036,46980355355,421272977267,3917446787884,

%U 37766791690501,376447420971545,3875957531387172,41149332371734371,449984429580538407,5061923434006018612,58517321729774406129

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

%C 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)).

%C 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

%H Vaclav Kotesovec, <a href="/A355337/b355337.txt">Table of n, a(n) for n = 0..568</a>

%H Thomas Scheuerle, <a href="/A355337/a355337.txt">An example for a(4) = 51 which explains the relation to polynomial roots</a>.

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

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

%F 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

%F 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

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

%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(exp(x) + x^2 - 1))) \\ _Michel Marcus_, Jun 29 2022

%o (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

%Y Cf. A000110, A277381, A316778, A355338.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Jun 29 2022