A216246 - OEIS

%I #8 Mar 14 2013 17:43:37

%S 1,1,1,2,7,20,62,228,867,3474,14864,67101,316028,1540782,7792810,

%T 40669011,218087867,1201778736,6792071574,39309770969,232718797430,

%U 1407837227275,8694994837673,54771751869237,351643945379956,2299330292987022,15302662859459784

%N O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(n*x^2)^n/n! * exp(-n*x*A(n*x^2)).

%C Compare to the LambertW identity:

%C Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

%e O.g.f.: A(x) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 20*x^5 + 62*x^6 + 228*x^7 +...

%e where

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

%e simplifies to a power series in x with integer coefficients.

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, k^k*x^k*subst(A, x, k*x^2)^k/k!*exp(-k*x*subst(A, x, k*x^2)+x*O(x^n)))); polcoeff(A, n)}

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

%Y Cf. A216247, A218672, A217900.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Mar 14 2013