A355291 - OEIS

%I #34 Jul 22 2022 02:26:53

%S 1,3,14,81,551,4266,36803,348543,3583484,39652659,468970211,

%T 5894584812,78366374813,1097537989671,16136598952718,248309032411485,

%U 3988468487017379,66715970326561170,1159712730763363991,20909709414253764819,390374806223071148084,7534929383736826736007

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

%C 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

%C 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

%H Seiichi Manyama, <a href="/A355291/b355291.txt">Table of n, a(n) for n = 0..500</a>

%H Vaclav Kotesovec, <a href="https://arxiv.org/abs/2207.10568">Asymptotics for a certain group of exponential generating functions</a>, arXiv:2207.10568 [math.CO], Jul 13 2022.

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

%F a(0) = 1; a(n) = Sum_{k=1..n} (1 + 2^k) * binomial(n-1,k-1) * a(n-k). - _Seiichi Manyama_, Jul 01 2022

%F a(n) ~ exp(exp(2*z) + exp(z) - 2 - n) * (n/z)^(n + 1/2) / sqrt(2*(1 + 2*z)*exp(2*z) + (1 + z)*exp(z)), where z = LambertW(n)/2 - 1/(1 + 2/LambertW(n) + 4 * n^(1/2) * (1 + LambertW(n)) / LambertW(n)^(3/2)). - _Vaclav Kotesovec_, Jul 03 2022

%F a(n) ~ 2^n * n^n / (sqrt(1 + LambertW(n)) * LambertW(n)^n * exp(n + 17/8 - n/LambertW(n) - sqrt(n/LambertW(n)))). - _Vaclav Kotesovec_, Jul 08 2022

%t nmax = 20; CoefficientList[Series[Exp[Exp[2*x] - 2 + Exp[x]], {x, 0, nmax}], x] * Range[0, nmax]!

%t Table[Sum[Binomial[n, k] * 2^k * BellB[k] * BellB[n-k], {k, 0, n}], {n, 0, 20}]

%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(exp(x)*(exp(x) + 1) - 2))) \\ _Michel Marcus_, Jun 27 2022

%Y Cf. A001861, A055882, A126390, A143405, A355379.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Jun 27 2022