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 A384269 #13 Jun 01 2025 04:41:19
%S A384269 1,1,2,6,16,49,154,513,1747,6078,21439,76607,276685,1008781,3707512,
%T A384269 13721086,51088860,191245836,719333008,2717229481,10303797518,
%U A384269 39208957744,149676496756,573037914270,2199735075908,8464921506665,32648239747059,126185248269567,488657718553676,1895790377527674
%N A384269 G.f. A(x) satisfies x = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 + x^n).
%C A384269 The g.f. utilizes the Jacobi triple product identity: Product_{n>=1} (1 - x^n/a)*(1 - x^(n-1)*a)*(1-x^n) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * a^n.
%H A384269 Paul D. Hanna, <a href="/A384269/b384269.txt">Table of n, a(n) for n = 0..630</a>
%F A384269 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas where theta_4(x) is a Jacobi elliptic function.
%F A384269 (1) x = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 + x^n).
%F A384269 (2) -x*A(x) = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 + x^n).
%F A384269 (3) x*theta_4(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 - x^n).
%F A384269 (4) -x*theta_4(x)*A(x) = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 - x^n).
%F A384269 (5.a) x*theta_4(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
%F A384269 (5.b) x*theta_4(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) / A(x)^n.
%F A384269 (6.a) -x*theta_4(x)*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * A(x)^n.
%F A384269 (6.b) -x*theta_4(x)*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
%F A384269 a(n) ~ c * d^n / n^(3/2), where d = 4.083846421711383847604673417919116998017... and c = 0.584432537831593677040363592052688856... - _Vaclav Kotesovec_, Jun 01 2025
%e A384269 G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 49*x^5 + 154*x^6 + 513*x^7 + 1747*x^8 + 6078*x^9 + 21439*x^10 + ...
%e A384269 RELATED SERIES.
%e A384269 1/A(x) = 1 - x - x^2 - 3*x^3 - 5*x^4 - 16*x^5 - 45*x^6 - 155*x^7 - 512*x^8 - 1763*x^9 - 6084*x^10 + ...
%e A384269 SPECIFIC VALUES.
%e A384269 A(exp(-Pi)) = 1.0474973549949421045732567080496722542518531011526934631...
%e A384269 where Sum_{n=-oo..+oo} (-1)^n * exp(-Pi*n*(n+1)/2) * A(exp(-Pi))^n = exp(-Pi) * (Pi/2)^(1/4) / gamma(3/4) = 0.03947933420376592813...
%e A384269 A(-exp(-Pi)) = 0.960086060200580366759936974556134222228793624085744940...
%e A384269 where Sum_{n=-oo..+oo} (-1)^(n*(n-1)/2) * exp(-Pi*n*(n+1)/2) * A(-exp(-Pi))^n = -exp(-Pi) * Pi^(1/4) / gamma(3/4) = -0.04694910513068872743...
%e A384269 A(t) = 2 at t = 0.24484187571695922418922496399796775078115821427621282...
%e A384269 A(t) = 7/4 at t = 0.239324355731620083092236573970947000576283799760943...
%e A384269 A(t) = 5/3 at t = 0.234439889083627870257298020352799276294012688627782...
%e A384269 A(t) = 3/2 at t = 0.217134571709901433113197085617818478214816713922905...
%e A384269 A(t) = 4/3 at t = 0.183806911401666173138177455971709388630788740531594...
%e A384269 A(t) = 5/4 at t = 0.157416870441717618165825450612923233287765184975643...
%e A384269 A(1/5) = 1.401449039483961854381757985869052435618161722574956...
%e A384269 A(1/6) = 1.276318946972284528693666572724710434062725174240448...
%e A384269 A(1/7) = 1.213287805382388838362413216213677242108560133326140...
%e A384269 A(1/8) = 1.174388177498186580244775740286834758637341200438483...
%e A384269 A(1/9) = 1.147764942051942680447238410304951699474657455354304...
%t A384269 (* Calculation of constants {d,c}: *) {1/r, -s*Log[r]/2* Sqrt[((s-1)*(-2*r*(s - 1) * QPochhammer[s, r] * Derivative[0, 1][QPochhammer][-1, r] + QPochhammer[-1, r]^2 * QPochhammer[s, r]^2 * Derivative[0, 1][QPochhammer][1/s, r] + 2*(s-1)* QPochhammer[-1, r] * (QPochhammer[s, r] - r*Derivative[0, 1][QPochhammer][s, r])))/ (Pi*QPochhammer[-1, r] * QPochhammer[s, r] * (-s*Log[ r]^2 + (s-1)^2 * (QPolyGamma[1, -Log[s]/Log[r], r] + QPolyGamma[1, Log[s]/Log[r], r])))]} /. FindRoot[{QPochhammer[-1, r] * QPochhammer[1/s, r] * QPochhammer[s, r] == 2*r*(1 - s), s*Log[r] + (s-1) * (QPolyGamma[0, Log[s]/Log[r], r] - QPolyGamma[0, Log[1/s]/Log[r], r]) == 0}, {r, 1/4}, {s, 2}, WorkingPrecision -> 120] (* _Vaclav Kotesovec_, Jun 01 2025 *)
%o A384269 (PARI) {a(n) = my(A=[1,1]); for(i=2,n, A=concat(A,0);
%o A384269 A[#A] = polcoef(x - prod(n=1,#A, (1 - x^n*Ser(A)) * (1 - x^(n-1)/Ser(A)) * (1 + x^n) ),#A-1); ); H=A; A[n+1]}
%o A384269 for(n=0,30,print1(a(n),", "))
%Y A384269 Cf. A356499, A384271.
%K A384269 nonn
%O A384269 0,3
%A A384269 _Paul D. Hanna_, May 25 2025