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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
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).
a(n) ~ c * d^n / n^(3/2), where d = 8.54148362320612002563896433934021488424489314523756456892173912667254... and c = 1.2437677914754786190190604348779334425700766084860016245397106832001... - Vaclav Kotesovec, Nov 16 2023
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, ...].
MATHEMATICA
CoefficientList[1/x * InverseSeries[Series[x/EllipticTheta[3, 0, x]^2, {x, 0, 25}], x], x] (* Vaclav Kotesovec, Nov 16 2023 *)
(* Calculation of constants {d, c}: *) {1/r, s/Sqrt[Pi*(1 + 4 * r^2 * s^(3/2) * Derivative[0, 0, 2][EllipticTheta][3, 0, r*s])]} /. FindRoot[{s == EllipticTheta[3, 0, r*s]^2, 2*r*Sqrt[s]*Derivative[0, 0, 1][EllipticTheta][3, 0, r*s] == 1}, {r, 1/8}, {s, 3/2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Nov 16 2023 *)
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 14 2012
STATUS
approved

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Last modified March 29 10:44 EDT 2024. Contains 371268 sequences. (Running on oeis4.)