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

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A177133 G.f. satisfies: A(A(x)) = Sum_{n>=1} A(x)^(n^2)/x^(n^2-n). 1
1, 1, 3, 12, 54, 261, 1344, 7380, 43099, 265739, 1717466, 11586670, 81422063, 594828360, 4508090145, 35380563603, 287130931064, 2406309163514, 20797551211656, 185158224231178, 1696132889163096, 15969702544475270 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Table of n, a(n) for n=1..22.

FORMULA

(1) G.f.: A(x) = Sum_{n>=1} x^(n^2)/G(x)^(n^2-n) where G(x) is the series reversion of A(x).

(2) Let q = A(x)/x, then g.f. A(x) satisfies the continued fraction:

A(A(x)) = -1 + 1/(1- q*x/(1- (q^3-q)*x/(1- q^5*x/(1- (q^7-q^3)*x/(1- q^9*x/(1- (q^11-q^5)*x/(1- q^13*x/(1- (q^15-q^7)*x/(1- ...)))))))))

due to an identity of a partial elliptic theta function.

(3) Let q = A(x)/x, then g.f. A(x) satisfies:

A(A(x)) = Sum_{n>=1} A(x)^n*Product_{k=1..n} (1-x*q^(4*k-3))/(1-x*q^(4*k-1)) due to a q-series identity.

EXAMPLE

G.f.: A(x) = x + x^2 + 3*x^3 + 12*x^4 + 54*x^5 + 261*x^6 + 1344*x^7 +...

where:

A(A(x)) = A(x) + A(x)^4/x^2 + A(x)^9/x^6 + A(x)^16/x^12 + A(x)^25/x^20 +...

Explicitly,

A(A(x)) = x + 2*x^2 + 8*x^3 + 40*x^4 + 222*x^5 + 1314*x^6 + 8172*x^7 + 53049*x^8 + 357905*x^9 + 2500608*x^10 +...

Related expansions:

A(x)^4/x^2 = x^2 + 4*x^3 + 18*x^4 + 88*x^5 + 451*x^6 + 2388*x^7 +...

A(x)^9/x^6 = x^3 + 9*x^4 + 63*x^5 + 408*x^6 + 2556*x^7 +...

A(x)^16/x^12 = x^4 + 16*x^5 + 168*x^6 + 1472*x^7 + 11684*x^8 +...

A(x)^25/x^20 = x^5 + 25*x^6 + 375*x^7 + 4400*x^8 + 44600*x^9 +...

A(x)^36/x^30 = x^6 + 36*x^7 + 738*x^8 + 11352*x^9 + 145899*x^10 +...

...

Let G(x) satisfy A(G(x)) = x, then

A(x) = x + x^4/G(x)^2 + x^9/G(x)^6 + x^16/G(x)^12 + x^25/G(x)^20 +...

where:

G(x) = x - x^2 - x^3 - 2*x^4 - 4*x^5 - 9*x^6 - 42*x^7 - 303*x^8 - 2000*x^9 - 11804*x^10 - 70275*x^11 - 459489*x^12 +...

Related expansions:

x^4/G(x)^2 = x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 40*x^6 + 116*x^7 +...

x^9/G(x)^6 = x^3 + 6*x^4 + 27*x^5 + 110*x^6 + 423*x^7 + 1566*x^8 +...

x^16/G(x)^12 = x^4 + 12*x^5 + 90*x^6 + 544*x^7 + 2895*x^8 +...

x^25/G(x)^20 = x^5 + 20*x^6 + 230*x^7 + 2000*x^8 + 14605*x^9 +...

x^36/G(x)^30 = x^6 + 30*x^7 + 495*x^8 + 5950*x^9 + 58245*x^10 +...

PROG

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

CROSSREFS

Sequence in context: A158826 A107264 A200740 * A186241 A193115 A270489

Adjacent sequences:  A177130 A177131 A177132 * A177134 A177135 A177136

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 10 2010

EXTENSIONS

Continued fraction formula corrected by Paul D. Hanna, Dec 13 2010

STATUS

approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 5 22:37 EST 2019. Contains 329782 sequences. (Running on oeis4.)