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!)
A212326 G.f. satisfies: A(x) = theta_3( x*A(x) )^2, where theta_3(x) is Jacobi's theta_3 function. 0
1, 4, 20, 112, 676, 4312, 28704, 197600, 1397060, 10090676, 74152456, 552666448, 4167528000, 31736182776, 243698432960, 1884809367456, 14668777816708, 114789815231560, 902661488046900, 7129068237647408, 56524456978032904, 449752267499647104 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..21.

FORMULA

G.f. A(x) satisfies:

(1) A(x) = (1 + 2*Sum_{n>=1} (x*A(x))^(n^2) )^2.

(2) A(x) =  1 + 4*Sum_{n>=1} (x*A(x))^n / (1 + (x*A(x))^(2*n)).

(3) A(x) = Product_{n>=1} (1 - (-x)^n*A(x)^n)^2 / (1 + (-x)^n*A(x)^n)^2.

(4) A( x/theta_3(x)^2 ) = theta_3(x)^2.

(5) A(x) = (1/x)*Series_Reversion(x/theta_3(x)^2), where theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).

a(n) = [x^n] theta_3(x)^(2*n+2) / (n+1).

EXAMPLE

G.f.: A(x) = 1 + 4*x + 20*x^2 + 112*x^3 + 676*x^4 + 4312*x^5 + 28704*x^6 +...

Given g.f. A(x), let q = x*A(x), then by a q-series identity:

A(x) = 1 + 4*q/(1+q^2) + 4*q^2/(1+q^4) + 4*q^3/(1+q^6) + 4*q^4/(1+q^8) +...

A(x) = (1 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + 2*q^25 +...)^2.

...

Illustrate a(n) = [x^n] theta_3(x)^(2*n+2) / (n+1) by the following table of coefficients in powers theta_3(x)^(2*n+2) for n>=0:

n=0: [(1), 4, 4, 0, 4, 8, 0, 0, 4, 4, 8, 0, 0, 8, 0, 0,...];

n=1: [1, (8), 24, 32, 24, 48, 96, 64, 24, 104, 144, 96, 96, 112,...];

n=2: [1, 12, (60), 160, 252, 312, 544, 960, 1020, 876, 1560, 2400,...];

n=3: [1, 16, 112, (448), 1136, 2016, 3136, 5504, 9328, 12112,...];

n=4: [1, 20, 180, 960, (3380), 8424, 16320, 28800, 52020, 88660,...];

n=5: [1, 24, 264, 1760, 7944, (25872), 64416, 133056, 253704,...];

n=6: [1, 28, 364, 2912, 16044, 64792, (200928), 503360, ...];

n=7: [1, 32, 480, 4480, 29152, 140736, 525952, (1580800), ...]; ...

where the coefficients in parenthesis form the initial terms of this sequence:

A = [1/1, 8/2, 60/3, 448/4, 3380/5, 25872/6, 200928/7, 1580800/8, ...].

PROG

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+4*sum(m=1, n, (x*A)^m/(1+(x*A+x*O(x^n))^(2*m)))); polcoeff(A, n)}

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

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

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A166952.

Sequence in context: A080609 A003645 A081085 * A192624 A209200 A294119

Adjacent sequences:  A212323 A212324 A212325 * A212327 A212328 A212329

KEYWORD

nonn

AUTHOR

Paul D. Hanna, May 14 2012

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 10 23:13 EST 2019. Contains 329909 sequences. (Running on oeis4.)