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A301363
G.f. A(x) satisfies: A(x) = 1/(1 - x*A(x)/(1 - 2*x*A(x)/(1 - 3*x*A(x)/(1 - 4*x*A(x)/(1 - ...))))), a continued fraction.
8
1, 1, 4, 25, 202, 1966, 22306, 289969, 4272934, 70792318, 1308702592, 26791202362, 602762346088, 14795609964448, 393567982759966, 11276489767853569, 346158428070229414, 11331678979354212886, 393967314482937530248, 14495027742943618066030, 562600190990455844759356
OFFSET
0,3
LINKS
FORMULA
G.f. A(x) satisfies: A(x) = 1 + x * (x*A(x)^3)' / (x*A(x))'. - Paul D. Hanna, Apr 01 2018
G.f. A(x) satisfies: A(x) = (1/x)*Series_Reversion(x/F(x)) where F(x) = A(x/F(x)) = Sum_{n>=0} (2*n)!/(n!*2^n)*x^n is an o.g.f. of A001147. - Paul D. Hanna, Apr 05 2018
G.f. A(x) satisfies: A(x) = Sum_{n>=0} (2*n)!/(n!*2^n) * x^n * A(x)^n. - Paul D. Hanna, Apr 09 2018
a(n) ~ 2^(n + 1/2) * n^n / exp(n - 1/2). - Vaclav Kotesovec, Jun 18 2019
EXAMPLE
G.f. A(x) = 1 + x + 4*x^2 + 25*x^3 + 202*x^4 + 1966*x^5 + 22306*x^6 + 289969*x^7 + 4272934*x^8 + ...
The g.f. also satisfies:
A(x) = 1 + x*A(x) + 3*x^2*A(x) + 15*x^3*A(x)^3 + 105*x^4*A(x)^4 + 945*x^5*A(x)^5 + 10395*x^6*A(x)^6 + ... + (2*n)!/(n!*2^n) * x^n * A(x)^n + ...
PROG
(PARI) /* Continued Fraction */
{a(n) = my(A=1, CF = 1+x +x*O(x^n)); for(i=1, n, A=CF; for(k=0, n, CF = 1/(1 - (n-k+1)*x*A*CF ) )); polcoeff(CF, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Differential Equation */
{a(n) = my(A=1); for(i=0, n, A = 1 + x*(x*A^3)'/(x*A +x^2*O(x^n))'); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Mar 19 2018
STATUS
approved