A360230 - OEIS

%I #10 Mar 14 2023 04:09:52

%S 1,1,3,19,109,921,8911,100003,1307769,18748369,307713691,5379610611,

%T 106277271013,2194176659689,50689643777319,1207518763542211,

%U 31940171681228401,862606920178886433,25708097594461923379,776354747057987797459,25741373454075987900381

%N a(n) = coefficient of x^n/n! in Sum_{n>=0} (1 + n*x + x^2)^n * x^n/n!.

%H Paul D. Hanna, <a href="/A360230/b360230.txt">Table of n, a(n) for n = 0..300</a>

%F E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! may be defined as follows.

%F (1) A(x) = Sum_{n>=0} (1 + n*x + x^2)^n * x^n/n!.

%F (2) A(x) = ( LambertW(-x^2)/(-x^2) )^(x + 1/x) / (1 + LambertW(-x^2)).

%F (3) A(x) = exp( -(x + 1/x) * LambertW(-x^2) ) / (1 + LambertW(-x^2)).

%F a(n) ~ (exp(exp(1/2) + exp(-1/2)) + (-1)^n*exp(- exp(1/2) - exp(-1/2))) * n^n / (sqrt(2) * exp(n/2)). - _Vaclav Kotesovec_, Mar 14 2023

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 109*x^4/4! + 921*x^5/5! + 8911*x^6/6! + 100003*x^7/7! + 1307769*x^8/8! + 18748369*x^9/9! + ...

%e such that

%e A(x) = 1 + (1 + x + x^2)*x + (1 + 2*x + x^2)^2*x^2/2! + (1 + 3*x + x^2)^3*x^3/3! + (1 + 4*x + x^2)^4*x^4/4! + (1 + 5*x + x^2)^5*x^5/5! + ... + (1 + n*x + x^2)^n * x^n/n! + ...

%e also

%e A(x) = W(x^2)^(x + 1/x) / (1 - x^2*W(x^2)), where W(x) = LambertW(-x)/(-x).

%o (PARI) {a(n) = n! * polcoeff( sum(m=0,n, (1 + m*x + x^2)^m * x^m/m! +x*O(x^n)),n)}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) {a(n) = n! * polcoeff( ( lambertw(-x^2 + O(x^(n+6)))/(-x^2) )^(x + 1/x) / (1 + lambertw(-x^2 + O(x^(n+6)))),n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A000169.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 19 2023