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.

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

Paul D. Hanna, Table of n, a(n) for n = 0..400

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

Cf. A001147, A088368, A302100, A302535, A302565.

Sequence in context: A254796 A347585 A088159 * A213192 A036242 A120955

Adjacent sequences: A301360 A301361 A301362 * A301364 A301365 A301366

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 19 2018

Status

approved