login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A145348 G.f. satisfies: A(x/A(x)^2) = 1 + x*A(x)^2. 5
1, 1, 4, 30, 312, 3965, 57824, 933998, 16346728, 305539046, 6037780164, 125227212342, 2711254371568, 61021656441091, 1423063422363676, 34297379607790288, 852463916004336464, 21812807282389353798 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

More generally, if g.f. A(x) satisfies: A(x/A(x)^k) = 1 + x*A(x)^m, then

A(x) = 1 + x*G(x)^(m+k) where G(x) = A(x*G(x)^k) and G(x/A(x)^k) = A(x);

thus a(n) = [x^(n-1)] ((m+k)/(m+k*n))*A(x)^(m+k*n) for n>=1 with a(0)=1.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..230

FORMULA

G.f.: A(x) = 1 + x*G(x)^4 where G(x) = A(x*G(x)^2) and A(x) = G(x/A(x)^2).

a(n) = [x^(n-1)] 2*A(x)^(2*n+2)/(n+1) for n>=1 with a(0)=1; i.e., a(n) equals the coefficient of x^(n-1) in 2*A(x)^(2*n+2)/(n+1) for n>=1 (see comment).

EXAMPLE

G.f.: A(x) = 1 + x + 4*x^2 + 30*x^3 + 312*x^4 + 3965*x^5 +...

A(x)^2 = 1 + 2*x + 9*x^2 + 68*x^3 + 700*x^4 + 8794*x^5 + 126974*x^6+..

A(x/A(x)^2) = 1 + x + 2*x^2 + 9*x^3 + 68*x^4 + 700*x^5 + 8794*x^6 +...

A(x) = 1 + x*G(x)^4 where G(x) = A(x*G(x)^2):

G(x) = 1 + x + 6*x^2 + 59*x^3 + 742*x^4 + 10877*x^5 + 177612*x^6 +...

G(x)^2 = 1 + 2*x + 13*x^2 + 130*x^3 + 1638*x^4 + 23946*x^5 +...

To illustrate the formula a(n) = [x^(n-1)] 2*A(x)^(2*n+2)/(n+1),

form a table of coefficients in A(x)^(2*n+2) as follows:

A^4: [(1), 4, 22, 172, 1753, 21612, 306348, ...];

A^6: [1, (6), 39, 320, 3267, 39756, 554595, ...];

A^8: [1, 8, (60), 520, 5366, 64816, 892308, ...];

A^10: [1, 10, 85, (780), 8190, 98702, 1344920,  ...];

A^12: [1, 12, 114, 1108, (11895), 143676, 1943488, ...];

A^14: [1, 14, 147, 1512, 16653, (202384), 2725541, ...]; ...

in which the main diagonal forms the initial terms of this sequence:

[2/2*(1), 2/3*(6), 2/4*(60), 2/5*(780), 2/6*(11895), 2/7*(202384), ...].

PROG

(PARI) {a(n)=local(F=1+x, G); for(i=0, n, G=serreverse(x/(F+x*O(x^n))^2)/x; F=1+x*G^2); polcoeff(F, n)}

(PARI) /* This sequence is generated when k=2, m=2: A(x/A(x)^k) = 1 + x*A(x)^m */

{a(n, k=2, m=2)=local(A=sum(i=0, n-1, a(i, k, m)*x^i)+x*O(x^n)); if(n==0, 1, polcoeff((m+k)/(m+k*n)*A^(m+k*n), n-1))}

for(n=0, 20, print1(a(n), ", "))

CROSSREFS

Cf. A145350, A147664, A120972.

Sequence in context: A052452 A088794 A239841 * A052574 A158834 A139086

Adjacent sequences:  A145345 A145346 A145347 * A145349 A145350 A145351

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 09 2008

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified November 20 17:42 EST 2014. Contains 249753 sequences.