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A245112 G.f. satisfies: A(x)^2 = 1 + 4*x*A(x)^5. 3
1, 2, 18, 224, 3230, 50688, 840420, 14483456, 256856886, 4656988160, 85929839996, 1608379269120, 30463651429484, 582796191989760, 11245047027447240, 218581150665277440, 4276257634911525670, 84135742205488791552, 1663738200769421021580, 33047906167191995678720 (list; graph; refs; listen; history; text; internal format)
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).

REFERENCES

Gi-Sang Cheon, S.-T. Jin, L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Available online 30 March 2015.

LINKS

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

FORMULA

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

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

Self convolution yields A214553.

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. A214553, A245113.

Sequence in context: A279045 A155666 A227934 * A260332 A254999 A024486

Adjacent sequences:  A245109 A245110 A245111 * A245113 A245114 A245115

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 31 2014

STATUS

approved

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Last modified October 23 00:39 EDT 2021. Contains 348211 sequences. (Running on oeis4.)