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 A384273 #13 Jul 01 2025 10:44:43
%S A384273 1,3,3,9,39,108,387,1581,5196,21573,82596,318279,1303146,5182389,
%T A384273 20919156,86577264,351929133,1462075095,6077250693,25277372124,
%U A384273 106131459906,445859648019,1878449392365,7955646845046,33707865532680,143344958486019,610977896794104,2608218534504888,11162376089875158
%N A384273 G.f. A(x) satisfies -3*x = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 + x^n).
%C A384273 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 A384273 Paul D. Hanna, <a href="/A384273/b384273.txt">Table of n, a(n) for n = 0..400</a>
%F A384273 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 A384273 (1) -3*x = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 + x^n).
%F A384273 (2) 3*x/A(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 + x^n).
%F A384273 (3) -3*x*theta_4(x) = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 - x^n).
%F A384273 (4) 3*x*theta_4(x)/A(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 - x^n).
%F A384273 (5.a) -3*x*theta_4(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
%F A384273 (5.b) -3*x*theta_4(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * A(x)^n.
%F A384273 (6.a) 3*x*theta_4(x)/A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) / A(x)^n.
%F A384273 (6.b) 3*x*theta_4(x)/A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
%F A384273 a(n) ~ c * d^n / n^(3/2), where d = 4.51389712142676799318649101918382471452597633164586338354... and c = 0.80137914461001211839391539872328025866773400116571811811... - _Vaclav Kotesovec_, Jun 30 2025
%e A384273 G.f.: A(x) = 1 + 3*x + 3*x^2 + 9*x^3 + 39*x^4 + 108*x^5 + 387*x^6 + 1581*x^7 + 5196*x^8 + 21573*x^9 + 82596*x^10 + 318279*x^11 + 1303146*x^12 + ...
%e A384273 where
%e A384273 -3*x = (1 - x/A(x))*(1 - A(x))*(1+x) * (1 - x^2/A(x))*(1 - x*A(x))*(1+x^2) * (1 - x^3/A(x))*(1 - x^2*A(x))*(1+x^3) * (1 - x^4/A(x))*(1 - x^3*A(x))*(1+x^4) * (1 - x^5/A(x))*(1 - x^4*A(x))*(1+x^5) * ...
%t A384273 (* Calculation of constants {d,c}: *) With[{k = 3}, Chop[{1/r, (1/Sqrt[2*Pi])*(-1 + s)* Sqrt[(s^2*(-r + s)*Log[r]*((r - s)*Log[1 - r] - r*Log[r] + (r - s)*(QPolyGamma[0, -1 + Log[s]/Log[r], r] + r*Log[r]*(Derivative[0, 1][QPochhammer][-1, r]/ QPochhammer[-1, r] + Derivative[0, 1][QPochhammer][1/s, r]/ QPochhammer[1/s, r] + Derivative[0, 1][QPochhammer][s/r, r]/ QPochhammer[s/r, r])))) / (-2*s*(r + r^2 - 3*r*s + s^3) * Log[r]^2 + 2*(-1 + s)*(-r + s)*(-r + s^2)* Log[r]*(QPolyGamma[0, -Log[s]/Log[r], r] - QPolyGamma[0, -1 + Log[s]/Log[r], r]) + (r - s)^2*(-1 + s)^2*((QPolyGamma[0, -Log[s]/Log[r], r] - QPolyGamma[0, -1 + Log[s]/Log[r], r]) * (Log[r] - QPolyGamma[0, -Log[s]/Log[r], r] + QPolyGamma[0, -1 + Log[s]/Log[r], r]) + QPolyGamma[1, -Log[s]/Log[r], r] + QPolyGamma[1, -1 + Log[s]/Log[r], r]))]} /. FindRoot[{2*k* r + (r*s*QPochhammer[-1, r]*QPochhammer[1/s, r] * QPochhammer[s/r, r])/((r - s)*(-1 + s)) == 0, (-r + s^2)*Log[r] + (r - s)*(-1 + s) * QPolyGamma[0, Log[1/s]/Log[r], r] - (r - s)*(-1 + s)*QPolyGamma[0, Log[s/r]/Log[r], r] == 0}, {r, 1/4}, {s, 2}, WorkingPrecision -> 120]]] (* _Vaclav Kotesovec_, Jun 30 2025 *)
%o A384273 (PARI) {a(n) = my(A=[1,3]); for(i=2,n, A=concat(A,0);
%o A384273 A[#A] = polcoef(3*x + prod(n=1,#A, (1 - x^n/Ser(A)) * (1 - x^(n-1)*Ser(A)) * (1 + x^n) ),#A-1); ); A[n+1]}
%o A384273 for(n=0,30,print1(a(n),", "))
%Y A384273 Cf. A356499, A384271, A384272.
%K A384273 nonn
%O A384273 0,2
%A A384273 _Paul D. Hanna_, Jun 29 2025