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A264232 G.f. satisfies: A(x)^2 = A( x^2/(1-6*x)^2 ). 3
1, 6, 39, 270, 1959, 14706, 113166, 887004, 7050837, 56672622, 459646488, 3756181248, 30893173038, 255509028612, 2123685458190, 17728918028172, 148590381782418, 1249839423702828, 10547139497197887, 89271390230559918, 757673193636234279, 6446893091203601298, 54983813851196942292, 469959567684908644440 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Radius of convergence is r = 1/9 where r = r^2/(1-6*r)^2 with A(r) = 1.

Compare to: C(x)^2 = C( x^2/(1-2*x)^2 ) where C(x) = (1-2*x-sqrt(1-4*x))/(2*x) is a g.f. of the Catalan numbers A000108.

LINKS

Table of n, a(n) for n=1..24.

FORMULA

G.f. satisfies:

(1) A(x) = -A( -x/(1-12*x) ).

(2) A(x^2) = A( x/(1+6*x) )^2 = A( -x/(1-6*x) )^2.

(3) A( x/(1+3*x)^2 ) = -A( -x/(1-3*x)^2 ), an odd function.

(4) A( x/(1+3*x)^2 )^2 = A( x^2/(1+9*x^2)^2 ), an even function.

(5) A( x/(1+4*x) ) = G(x) = Sum(n>=1} A264224(n)*x^n where G(x)^2 = G( x^2/(1-4*x) ).

(6) A( x/(1+8*x) ) = -G(-x) = Sum(n>=1} (-1)^(n-1) * A264224(n)*x^n where G(x)^2 = G( x^2/(1-4*x) ).

Sum_{k=0..n} binomial(n,k) * (-6)^(n-k) * a(k+1) = 0 for odd n.

Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * a(k+1) = A264224(n+1) for n>=0.

Sum_{k=0..n} binomial(n,k) * (-8)^(n-k) * a(k+1) = (-1)^n * A264224(n+1) for n>=0.

EXAMPLE

G.f.: A(x) = x + 6*x^2 + 39*x^3 + 270*x^4 + 1959*x^5 + 14706*x^6 + 113166*x^7 + 887004*x^8 + 7050837*x^9 + 56672622*x^10 + 459646488*x^11 + 3756181248*x^12 +...

where A( x^2/(1-6*x)^2 ) = A(x)^2,

A( x^2/(1-6*x)^2 ) = x^2 + 12*x^3 + 114*x^4 + 1008*x^5 + 8679*x^6 + 73980*x^7 + 628506*x^8 + 5336928*x^9 + 45351591*x^10 + 385869348*x^11 + 3287962710*x^12 +...

Also, A( x/(1+6*x) ) = A(x^2)^(1/2),

A( x/(1+6*x) ) = x + 3*x^3 + 15*x^5 + 90*x^7 + 597*x^9 + 4212*x^11 + 30942*x^13 + 233766*x^15 + 1802706*x^17 + 14122359*x^19 + 112033791*x^21 + 898024320*x^23 +...

Let B(x) = x/Series_Reversion( A(x) ), so that A(x) = x*B(A(x)), then

B(x) = 1 + 6*x + 3*x^2 - 3*x^4 + 9*x^6 - 33*x^8 + 126*x^10 - 513*x^12 + 2214*x^14 - 9876*x^16 + 45045*x^18 - 209493*x^20 + 990198*x^22 +...+ A264412(n)*x^(2*n) +...

such that B(x) = F(x^2) + 6*x = F(x)^2 where F(x) is the g.f. of A264412.

PROG

(PARI) {a(n) = my(A=x); for(i=1, #binary(n+1), A = sqrt( subst(A, x, x^2/(1-6*x +x*O(x^n))^2) ) ); polcoeff(A, n)}

for(n=1, 30, print1(a(n), ", "))

CROSSREFS

Cf. A264412, A264224, A264233, A260650.

Sequence in context: A052392 A199491 A147961 * A068765 A349531 A006633

Adjacent sequences:  A264229 A264230 A264231 * A264233 A264234 A264235

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 16 2015

STATUS

approved

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Last modified August 9 15:33 EDT 2022. Contains 356026 sequences. (Running on oeis4.)