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A216879
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G.f. satisfies: A(x) = sqrt( theta_3(x*A(x)) / theta_4(x*A(x)) ).
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0
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1, 2, 6, 24, 110, 540, 2772, 14704, 79974, 443594, 2499640, 14269320, 82346004, 479604748, 2815557264, 16643093712, 98974828886, 591742372068, 3554708076858, 21444913596408, 129870710693976, 789237890852160, 4811481299622276, 29417496447990096, 180337119342194820
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OFFSET
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0,2
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COMMENTS
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Here theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2) and theta_4(x) = 1 + 2*Sum_{n>=1} (-x)^(n^2) are Jacobi theta functions.
The radius of convergence r of g.f. A(x) is given by
r = 0.15335406881552899483841215094726329935743212998703... with
A(r) = 2.14877235788136654366723937779352044712735012012453...
such that G(y) = y*G'(y) = A(r) at y = r*A(r) = 0.3295229840394455820300...
where G(x) = sqrt(theta_3(x)/theta_4(x)).
Conjectured to be the number of permutations of length n+1 avoiding the partially ordered pattern (POP) {5>1, 1>2, 1>4} of length 5. That is, conjectured to be the number of length n+1 permutations having no subsequences of length 5 in which the fifth element is larger than the first element, which in turn is larger than the second and fourth elements. - Sergey Kitaev, Dec 13 2020
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LINKS
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FORMULA
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G.f. satisfies the identities:
(1) A(x) = 1 / A(-x*A(x)^2).
(2) A(x) = eta(-x*A(x))^2 * eta(x^4*A(x)^4) / eta(x^2*A(x)^2)^3.
(3) A(x) = exp( 2*Sum_{n>=1} sigma(2*n-1) * (x*A(x))^(2*n-1) / (2*n-1) ).
(4) A(x) = 1 / Product_{n>=1} (1 + (x*A(x))^(2*n)) * (1 - (x*A(x))^(2*n-1))^2.
(5) A(x) = Product_{n>=1} (1 + (x*A(x))^(2*n-1)) * (1 + (x*A(x))^n).
(6) A(x) = Product_{n>=1} (1 + (x*A(x))^(2*n-1)) / (1 - (x*A(x))^(2*n-1)).
(7) A(x) = Product_{n>=1} (1 - (x*A(x))^(4*n-2)) / ((1 - (x*A(x))^(4*n-1))*(1 - (x*A(x))^(4*n-3)))^2.
(8) A(x) = 1/(1 - 2*q/(1+q - q^2*(1-q^2)/(1+q^3 - q^3*(1-q^4)/(1+q^5 - q^4*(1-q^6)/(1+q^7 - ...))))), a continued fraction, where q = x*A(x).
(9) A(x) = (1/x)*Series_Reversion( x*sqrt(theta_4(x)/theta_3(x)) ).
(10) A(x/G(x)) = G(x) where G(x) = sqrt(theta_3(x)/theta_4(x)) is the g.f. of A080054.
Special value: A(exp(-Pi)/2^(1/8)) = 2^(1/8).
a(n) = [x^n] ( theta_3(x) / theta_4(x) )^((n+1)/2) / (n+1).
a(n) ~ c * d^n / n^(3/2), where d = 6.52085730573545526010335599231748172235904166255252115709479430152403... and c = 0.6370998492207183978277090515469899143891211207560886906399176320450... - Vaclav Kotesovec, Nov 16 2023
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 6*x^2 + 24*x^3 + 110*x^4 + 540*x^5 + 2772*x^6 +...
such that, by definition, the g.f. satisfies:
A(x) = sqrt( (1 + 2*Sum_{n>=1} (x*A(x))^(n^2) ) / (1 + 2*Sum_{n>=1} (-x*A(x))^(n^2) ) ).
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MATHEMATICA
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InverseSeries[x Sqrt[EllipticTheta[4, 0, x]/EllipticTheta[3, 0, x]] + O[x]^26] // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Oct 01 2019 *)
(* Calculation of constants {d, c}: *) {1/r, s*Sqrt[EllipticTheta[3, 0, r*s] / (Pi*(6*EllipticTheta[3, 0, r*s] - r*s*(4*Derivative[0, 0, 1][EllipticTheta][3, 0, r*s] - r*s*Derivative[0, 0, 2][EllipticTheta][3, 0, r*s] + r*s^3*Derivative[0, 0, 2][EllipticTheta][4, 0, r*s])))]} /. FindRoot[{EllipticTheta[3, 0, r*s]/EllipticTheta[4, 0, r*s] == s^2, (r*(Derivative[0, 0, 1][EllipticTheta][3, 0, r*s] - s^2*Derivative[0, 0, 1][EllipticTheta][4, 0, r*s])) / (2*s*EllipticTheta[4, 0, r*s]) == 1}, {r, 1/6}, {s, 3/2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Nov 16 2023 *)
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PROG
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(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=sqrt((1+2*sum(m=1, sqrtint(n)+1, (x*A)^(m^2)))/(1+2*sum(m=1, sqrtint(n)+1, (-x*A)^(m^2))))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=eta(-x*A)^2*eta(x^4*A^4)/eta(x^2*A^2)^3); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(2*sum(n=1, n, sigma(2*n-1)*(x*A)^(2*n-1)/(2*n-1)))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/prod(m=1, n, (1+(x*A)^(2*m))*(1-(x*A)^(2*m-1))^2)); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=prod(m=1, n, (1+(x*A)^(2*m-1))*(1+(x*A)^m))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=prod(m=1, n, (1+(x*A)^(2*m-1))/(1-(x*A)^(2*m-1)))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=prod(m=1, n, (1-(x*A)^(4*m-2))/((1-(x*A)^(4*m-1))*(1-(x*A)^(4*m-3)))^2)); polcoeff(A, n)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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