This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A216879 #45 Jul 22 2024 15:50:34
%S A216879 1,2,6,24,110,540,2772,14704,79974,443594,2499640,14269320,82346004,
%T A216879 479604748,2815557264,16643093712,98974828886,591742372068,
%U A216879 3554708076858,21444913596408,129870710693976,789237890852160,4811481299622276,29417496447990096,180337119342194820
%N A216879 G.f. satisfies: A(x) = sqrt( theta_3(x*A(x)) / theta_4(x*A(x)) ).
%C A216879 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.
%C A216879 The radius of convergence r of g.f. A(x) is given by
%C A216879 r = 0.15335406881552899483841215094726329935743212998703... with
%C A216879 A(r) = 2.14877235788136654366723937779352044712735012012453...
%C A216879 such that G(y) = y*G'(y) = A(r) at y = r*A(r) = 0.3295229840394455820300...
%C A216879 where G(x) = sqrt(theta_3(x)/theta_4(x)).
%C A216879 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
%C A216879 The conjecture was disproven. The numbers are actually A366706, which matches the first 9 entries. - _Christian Bean_, Jul 22 2024
%H A216879 Christian Bean, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, <a href="https://doi.org/10.37236/12686">Permutations avoiding bipartite partially ordered patterns have a regular insertion encoding</a>, The Electronic Journal of Combinatorics, Volume 31, Issue 3 (2024); <a href="https://arxiv.org/abs/2312.07716">arXiv preprint</a>, arXiv:2312.07716 [math.CO], 2023.
%H A216879 Alice L. L. Gao and Sergey Kitaev, <a href="https://arxiv.org/abs/1903.08946">On partially ordered patterns of length 4 and 5 in permutations</a>, arXiv:1903.08946 [math.CO], 2019.
%H A216879 Alice L. L. Gao and Sergey Kitaev, <a href="https://doi.org/10.37236/8605">On partially ordered patterns of length 4 and 5 in permutations</a>, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
%F A216879 G.f. satisfies the identities:
%F A216879 (1) A(x) = 1 / A(-x*A(x)^2).
%F A216879 (2) A(x) = eta(-x*A(x))^2 * eta(x^4*A(x)^4) / eta(x^2*A(x)^2)^3.
%F A216879 (3) A(x) = exp( 2*Sum_{n>=1} sigma(2*n-1) * (x*A(x))^(2*n-1) / (2*n-1) ).
%F A216879 (4) A(x) = 1 / Product_{n>=1} (1 + (x*A(x))^(2*n)) * (1 - (x*A(x))^(2*n-1))^2.
%F A216879 (5) A(x) = Product_{n>=1} (1 + (x*A(x))^(2*n-1)) * (1 + (x*A(x))^n).
%F A216879 (6) A(x) = Product_{n>=1} (1 + (x*A(x))^(2*n-1)) / (1 - (x*A(x))^(2*n-1)).
%F A216879 (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.
%F A216879 (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).
%F A216879 (9) A(x) = (1/x)*Series_Reversion( x*sqrt(theta_4(x)/theta_3(x)) ).
%F A216879 (10) A(x/G(x)) = G(x) where G(x) = sqrt(theta_3(x)/theta_4(x)) is the g.f. of A080054.
%F A216879 Special value: A(exp(-Pi)/2^(1/8)) = 2^(1/8).
%F A216879 a(n) = [x^n] ( theta_3(x) / theta_4(x) )^((n+1)/2) / (n+1).
%F A216879 a(n) ~ c * d^n / n^(3/2), where d = 6.52085730573545526010335599231748172235904166255252115709479430152403... and c = 0.6370998492207183978277090515469899143891211207560886906399176320450... - _Vaclav Kotesovec_, Nov 16 2023
%e A216879 G.f.: A(x) = 1 + 2*x + 6*x^2 + 24*x^3 + 110*x^4 + 540*x^5 + 2772*x^6 +...
%e A216879 such that, by definition, the g.f. satisfies:
%e A216879 A(x) = sqrt( (1 + 2*Sum_{n>=1} (x*A(x))^(n^2) ) / (1 + 2*Sum_{n>=1} (-x*A(x))^(n^2) ) ).
%t A216879 InverseSeries[x Sqrt[EllipticTheta[4, 0, x]/EllipticTheta[3, 0, x]] + O[x]^26] // CoefficientList[#, x]& // Rest (* _Jean-François Alcover_, Oct 01 2019 *)
%t A216879 (* 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 *)
%o A216879 (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)}
%o A216879 for(n=0,20,print1(a(n),", "))
%o A216879 (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)}
%o A216879 (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)}
%o A216879 (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)}
%o A216879 (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)}
%o A216879 (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)}
%o A216879 (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)}
%Y A216879 Cf. A080054.
%K A216879 nonn
%O A216879 0,2
%A A216879 _Paul D. Hanna_, Sep 18 2012