A245112 G.f. A(x) satisfies: A(x)^2 = 1 + 4*x*A(x)^5.

1, 2, 18, 224, 3230, 50688, 840420, 14483456, 256856886, 4656988160, 85929839996, 1608379269120, 30463651429484, 582796191989760, 11245047027447240, 218581150665277440, 4276257634911525670, 84135742205488791552, 1663738200769421021580, 33047906167191995678720

Offset

0,2

Comments

Radius of convergence of g.f. A(x) is r = (3/5)^(5/2) / 6 where A(r) = sqrt(5/3).

Links

Table of n, a(n) for n=0..19.

Gi-Sang Cheon, S.-T. Jin, and L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Volume 491, 15 February 2016, Pages 123-137.

Formula

a(n) = 4^n * binomial((5*n - 1)/2, n) / (3*n + 1).

G.f. A(x) satisfies: A(x) = sqrt(1 + 4*x^2*A(x)^8) + 2*x*A(x)^4.

Self convolution yields A214553.

G.f. A(x) = 1/x * series reversion of x*sqrt(1 - 4*x*C(4*x)), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the generating function of the Catalan numbers A000108. See A024491. - Peter Bala, Mar 27 2023

Example

G.f.: A(x) =  = 1 + 2*x + 18*x^2 + 224*x^3 + 3230*x^4 + 50688*x^5 +...
where A(x)^2 = 1 + 4*x*A(x)^5:
A(x)^2 = 1 + 4*x + 40*x^2 + 520*x^3 + 7680*x^4 + 122360*x^5 +...
A(x)^5 = 1 + 10*x + 130*x^2 + 1920*x^3 + 30590*x^4 + 512512*x^5 +...
Related series:
A(x)^4 = 1 + 8*x + 96*x^2 + 1360*x^3 + 21120*x^4 + 347760*x^5 +...
A(x)^8 = 1 + 16*x + 256*x^2 + 4256*x^3 + 73216*x^4 + 1294560*x^5 +...
where A(x) = sqrt(1 + 4*x^2*A(x)^8) + 2*x*A(x)^4.

Prog

(PARI) /* From A(x)^2 = 1 + 4*x*A(x)^5 : */
{a(n) = local(A=1+x); for(i=1, n, A=sqrt(1 + 4*x*A^5 +x*O(x^n))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = 4^n * binomial((5*n - 1)/2, n) / (3*n + 1)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* From A(x) = sqrt(1 + 4*x^2*A(x)^8) + 2*x*A(x)^4 : */
{a(n) = local(A=1+x); for(i=1, n, A = sqrt(1 + 4*x^2*A^8 +x*O(x^n)) + 2*x*A^4); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A000108, A024491, A214553, A245113.

Sequence in context: A227934 A349652 A364825 * A260332 A254999 A364167

Adjacent sequences: A245109 A245110 A245111 * A245113 A245114 A245115

Keyword

nonn,easy

Author

Paul D. Hanna, Jul 31 2014

Status

approved