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 A192623 #15 Jun 30 2025 05:46:47
%S A192623 1,2,4,16,70,336,1720,9152,50140,280882,1601496,9263424,54224312,
%T A192623 320611152,1912003536,11487287872,69463274022,422440713680,
%U A192623 2582081184572,15853795192704,97736576247976,604744065493936,3754311394271208,23377930236777152
%N A192623 G.f. satisfies: A(x) = Product_{n>=0} (1 + x^(n+1)*A(x)^(2*n))/(1 - x^(n+1)*A(x)^(2*n)).
%C A192623 Related q-series (Heine) identity:
%C A192623 1 + Sum_{n>=1} x^n*Product_{k=0..n-1} (y+q^k)*(z+q^k)/((1-x*q^k)*(1-q^(k+1)) = Product_{n>=0} (1+x*y*q^n)*(1+x*z*q^n)/((1-x*q^n)*(1-x*y*z*q^n)); here q=x*A(x)^2, x=x, y=1, z=0.
%H A192623 Vaclav Kotesovec, <a href="/A192623/b192623.txt">Table of n, a(n) for n = 0..230</a>
%F A192623 G.f. satisfies:
%F A192623 (1) A(x) = 1 + Sum_{n>=1} x^(n*(n+1)/2)*A(x)^(n*(n-1)) * Product_{k=0..n-1} (1 + x^k*A(x)^(2*k))/((1 - x^(k+1)*A(x)^(2*k))*(1 - x^(k+1)*A(x)^(2*k+2))), due to the Heine identity.
%F A192623 (2) A(x)^2 = 1 + Sum_{n>=1} x^n * Product_{k=0..n-1} (1 + x^k*A(x)^(2*k))^2/((1 - x^(k+1)*A(x)^(2*k))*(1 - x^(k+1)*A(x)^(2*k+2)), due to the Heine identity.
%F A192623 Self-convolution yields A192622.
%F A192623 a(n) ~ c * d^n / n^(3/2), where d = 6.6513304693895827154026... and c = 0.31031758618989325658... - _Vaclav Kotesovec_, Jun 30 2025
%e A192623 G.f.: A(x) = 1 + 2*x + 4*x^2 + 16*x^3 + 70*x^4 + 336*x^5 + 1720*x^6 +...
%e A192623 The g.f. A = A(x) satisfies the following relations:
%e A192623 (0) A = (1+x)/(1-x) * (1+x^2*A^2)/(1-x^2*A^2) * (1+x^3*A^4)/(1-x^3*A^4) * (1+x^4*A^6)/(1-x^4*A^6)*...
%e A192623 (1) A = 1 + 2*x/((1-x)*(1-x*A^2)) + 2*x^3*A^2*(1+x*A^2)/((1-x)*(1-x*A^2)*(1-x^2*A^2)*(1-x^2*A^4)) + 2*x^6*A^6*(1+x*A^2)*(1+x^2*A^4)/((1-x)*(1-x*A^2)*(1-x^2*A^2)*(1-x^2*A^4)*(1-x^3*A^4)*(1-x^3*A^6)) +...
%e A192623 (2) A^2 = 1 + 4*x/((1-x)*(1-x*A^2)) + 4*x^2*(1+x*A^2)^2/((1-x)*(1-x*A^2)*(1-x^2*A^2)*(1-x^2*A^4)) + 4*x^3*(1+x*A^2)^2*(1+x^2*A^4)^2/((1-x)*(1-x*A^2)*(1-x^2*A^2)*(1-x^2*A^4)*(1-x^3*A^4)*(1-x^3*A^6)) +...
%t A192623 (* Calculation of constants {d,c}: *) Chop[{1/r, (s*Sqrt[(QPochhammer[r, r*s^2]*(Log[r*s^2] - 2*QPolyGamma[0, Log[-r]/Log[r*s^2], r*s^2] + 2*QPolyGamma[0, Log[r]/Log[r*s^2], r*s^2]))/(Log[ r*s^2]*(QPochhammer[r, r*s^2] - 4*r*s^2*(Derivative[0, 1][QPochhammer][r, r*s^2] + r*s*(-Derivative[0, 2][QPochhammer][-r, r*s^2] + s*Derivative[0, 2][QPochhammer][r, r*s^2]))))])/(2* Sqrt[Pi])} /. FindRoot[{s == QPochhammer[-r, r*s^2]/QPochhammer[r, r*s^2], QPochhammer[r, r*s^2] + 2*r*s^2*Derivative[0, 1][QPochhammer][r, r*s^2] == 2*r*s*Derivative[0, 1][QPochhammer][-r, r*s^2]}, {r, 1/6}, {s, 3/2}, WorkingPrecision -> 120]] (* _Vaclav Kotesovec_, Jun 30 2025 *)
%o A192623 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=prod(k=0,n,(1+x^(k+1)*A^(2*k))/(1-x^(k+1)*(A+x*O(x^n))^(2*k))));polcoeff(A,n)}
%o A192623 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^(m*(m+1)/2)*A^(m*(m-1))*prod(k=0,m-1,(1+x^k*A^(2*k))/((1-x^(k+1)*A^(2*k))*(1-x^(k+1)*A^(2*k+2) +x*O(x^n))))));polcoeff(A,n)}
%o A192623 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sqrt(1+sum(m=1,n,x^m*prod(k=0,m-1,(1+x^k*A^(2*k))^2/((1-x^(k+1)*A^(2*k) +x*O(x^n))*(1-x^(k+1)*A^(2*k+2)))))));polcoeff(A,n)}
%Y A192623 Cf. A192622 (g.f. A(x)^2), A192621, A192624.
%K A192623 nonn
%O A192623 0,2
%A A192623 _Paul D. Hanna_, Jul 06 2011