A219220 O.g.f. satisfies: A(x) = Sum_{n>=0} (n+1)^(n-1) * (n*x)^n * A(n*x)^(2*n)/n! * exp(-(n+1)*n*x*A(n*x)^2).

1, 1, 6, 91, 2306, 86576, 4570570, 333164243, 33547502582, 4704103190166, 925622587155708, 256758944391842662, 100693326907873920440, 55964816627849652514434, 44167198051129910003931850, 49561249392391287991062025027, 79164926515567602205248823277126

Offset

0,3

Comments

Compare the g.f. to the LambertW identity:

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

Links

Table of n, a(n) for n=0..16.

Example

 O.g.f.: A(x) = 1 + x + 6*x^2 + 91*x^3 + 2306*x^4 + 86576*x^5 +...
where
A(x) = 1 + 2^0*1^1*x*A(x)^2*exp(-2*1*x*A(x)^2) + 3^1*2^2*x^2*A(2*x)^4*exp(-3*2*x*A(2*x)^2)/2! + 4^2*3^3*x^3*A(3*x)^6*exp(-4*3*x*A(3*x)^2)/3! + 5^3*4^4*x^4*A(4*x)^8*exp(-5*4*x*A(4*x)^2)/4! + 6^4*5^5*x^5*A(5*x)^10*exp(-6*5*x*A(5*x)^2)/5! +...
simplifies to a power series in x with integer coefficients.

Prog

 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, (k+1)^(k-1)*k^k*x^k*subst(A^2, x, k*x)^k/k!*exp(-(k+1)*k*x*subst(A^2, x, k*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A218102, A217900.

Sequence in context: A095864 A246155 A349716 * A006151 A005327 A182263

Adjacent sequences: A219217 A219218 A219219 * A219221 A219222 A219223

Keyword

nonn

Author

Paul D. Hanna, Nov 16 2012

Status

approved