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 A192405 #5 Mar 30 2012 18:37:27
%S A192405 1,0,1,2,4,11,33,99,310,1016,3413,11682,40751,144476,519013,1886311,
%T A192405 6928012,25684055,96020957,361742039,1372442092,5241062187,
%U A192405 20136335035,77806111700,302259125863,1180207733657,4630733662020,18254415188073,72283753111667
%N A192405 G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^(n+1) * A(x)^n/(1 - x*A(x)^(2*n)).
%C A192405 Related q-series identity:
%C A192405 Sum_{n>=1} z^n*y*q^n/(1-y*q^(2*n)) = Sum_{n>=1} y^n*z*q^(2*n-1)/(1-z*q^(2*n-1)); here q=A(x), y=x, z=x.
%F A192405 G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^(n+1)*A(x)^(2*n-1)/(1 - x*A(x)^(2*n-1)).
%F A192405 G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^(n+1)*A(x)^(n*(n+1)/2) * Sum_{k=0..n-1} A(x)^(-k*(k+1)/2).
%F A192405 Equals the antidiagonal sums of square array A192404.
%e A192405 G.f.: A(x) = 1 + x^2 + 2*x^3 + 4*x^4 + 11*x^5 + 33*x^6 + 99*x^7 +...
%e A192405 which satisfies the following relations:
%e A192405 A(x) = 1 + x^2*A(x)/(1-x*A(x)^2) + x^3*A(x)^2/(1-x*A(x)^4) + x^4*A(x)^3/(1-x*A(x)^6) +...
%e A192405 A(x) = 1 + x^2*A(x)/(1-x*A(x)) + x^3*A(x)^3/(1-x*A(x)^3) + x^4*A(x)^5/(1-x*A(x)^5) +...
%e A192405 A(x) = 1 + x^2*A(x) + x^3*A(x)^3*(1 + 1/A(x)) + x^4*A(x)^6*(1 + 1/A(x) + 1/A(x)^3) + x^5*A(x)^10*(1 + 1/A(x) + 1/A(x)^3 + 1/A(x)^6) +...
%o A192405 (PARI) {a(n)=local(A=1+x^2);for(i=1,n,A=1+x*sum(m=1,n,x^m*A^m/(1-x*A^(2*m)+x*O(x^n))));polcoeff(A,n)}
%o A192405 (PARI) {a(n)=local(A=1+x^2);for(i=1,n,A=1+x*sum(m=1,n,x^m*A^(2*m-1)/(1-x*A^(2*m-1)+x*O(x^n))));polcoeff(A,n)}
%o A192405 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^(m+1)*A^(m*(m+1)/2)*sum(k=0,m-1,(A+x*O(x^n))^(-k*(k+1)/2) ) ) );polcoeff(A,n)}
%Y A192405 Cf. A192404, A192406, A192399, A192401.
%K A192405 nonn
%O A192405 0,4
%A A192405 _Paul D. Hanna_, Jun 30 2011