A317351 G.f. satisfies: A(x) = Sum_{n>=0} ( (1+x)^(n+1) - A(x) )^n / (2 - (1+x)^n*A(x))^(n+1).

1, 2, 6, 16, 154, 4584, 130464, 3816304, 123180090, 4422532004, 175136909492, 7585703878304, 356923128965592, 18139717839708536, 990827454743868120, 57910782633622271952, 3607453763547725076028, 238660376246383050751764, 16714929289459273370819900, 1235688614706272361317140840, 96170725583233854961162923028

Offset

0,2

Comments

G.f. A(x) = G(log(1+x)), where G(x) is the e.g.f. of A317356.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

G.f. A(x) satisfies:

(1) A(x) = Sum_{n>=0} ( (1+x)^(n+1) - A(x) )^n / (2 - (1+x)^n*A(x))^(n+1),

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

a(n) ~ c * d^n * n! / sqrt(n), where d = A317904 = 3.956184203026... and c = 0.23137523927... - Vaclav Kotesovec, Aug 07 2018

Example

G.f.: A(x) = 1 + 2*x + 6*x^2 + 16*x^3 + 154*x^4 + 4584*x^5 + 130464*x^6 + 3816304*x^7 + 123180090*x^8 + 4422532004*x^9 + 175136909492*x^10 + ...
such that A = A(x) satisfies
A(x) = 1/(2 - A)  +  ((1+x)^2 - A)/(2 - (1+x)*A)^2  +  ((1+x)^3 - A)^2/(2 - (1+x)^2*A)^3  +  ((1+x)^4 - A)^3/(2 - (1+x)^3*A)^4  +  ((1+x)^5 - A)^4/(2 - (1+x)^4*A)^5  +  ((1+x)^6 - A)^5/(2 - (1+x)^5*A)^6 + ...
Also,
A(x) = 1/(2 + A)  +  ((1+x)^2 + A)/(2 + (1+x)*A)^2  +  ((1+x)^3 + A)^2/(2 + (1+x)^2*A)^3  +  ((1+x)^4 + A)^3/(2 + (1+x)^3*A)^4  +  ((1+x)^5 + A)^4/(2 + (1+x)^4*A)^5  +  ((1+x)^6 + A)^5/(2 + (1+x)^5*A)^6 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A = Vec( sum(m=0, #A, ( (1+x)^(m+1) - Ser(A) )^m  / (2 - (1+x)^m*Ser(A))^(m+1) ) ) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A317350, A317356.

Sequence in context: A341838 A325790 A144690 * A296108 A118305 A139629

Adjacent sequences: A317348 A317349 A317350 * A317352 A317353 A317354

Keyword

nonn

Author

Paul D. Hanna, Aug 02 2018

Status

approved