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 A145348 #10 Aug 10 2014 06:53:33
%S A145348 1,1,4,30,312,3965,57824,933998,16346728,305539046,6037780164,
%T A145348 125227212342,2711254371568,61021656441091,1423063422363676,
%U A145348 34297379607790288,852463916004336464,21812807282389353798
%N A145348 G.f. satisfies: A(x/A(x)^2) = 1 + x*A(x)^2.
%C A145348 More generally, if g.f. A(x) satisfies: A(x/A(x)^k) = 1 + x*A(x)^m, then
%C A145348 A(x) = 1 + x*G(x)^(m+k) where G(x) = A(x*G(x)^k) and G(x/A(x)^k) = A(x);
%C A145348 thus a(n) = [x^(n-1)] ((m+k)/(m+k*n))*A(x)^(m+k*n) for n>=1 with a(0)=1.
%H A145348 Vaclav Kotesovec, <a href="/A145348/b145348.txt">Table of n, a(n) for n = 0..230</a>
%F A145348 G.f.: A(x) = 1 + x*G(x)^4 where G(x) = A(x*G(x)^2) and A(x) = G(x/A(x)^2).
%F A145348 a(n) = [x^(n-1)] 2*A(x)^(2*n+2)/(n+1) for n>=1 with a(0)=1; i.e., a(n) equals the coefficient of x^(n-1) in 2*A(x)^(2*n+2)/(n+1) for n>=1 (see comment).
%e A145348 G.f.: A(x) = 1 + x + 4*x^2 + 30*x^3 + 312*x^4 + 3965*x^5 +...
%e A145348 A(x)^2 = 1 + 2*x + 9*x^2 + 68*x^3 + 700*x^4 + 8794*x^5 + 126974*x^6+..
%e A145348 A(x/A(x)^2) = 1 + x + 2*x^2 + 9*x^3 + 68*x^4 + 700*x^5 + 8794*x^6 +...
%e A145348 A(x) = 1 + x*G(x)^4 where G(x) = A(x*G(x)^2):
%e A145348 G(x) = 1 + x + 6*x^2 + 59*x^3 + 742*x^4 + 10877*x^5 + 177612*x^6 +...
%e A145348 G(x)^2 = 1 + 2*x + 13*x^2 + 130*x^3 + 1638*x^4 + 23946*x^5 +...
%e A145348 To illustrate the formula a(n) = [x^(n-1)] 2*A(x)^(2*n+2)/(n+1),
%e A145348 form a table of coefficients in A(x)^(2*n+2) as follows:
%e A145348 A^4: [(1), 4, 22, 172, 1753, 21612, 306348, ...];
%e A145348 A^6: [1, (6), 39, 320, 3267, 39756, 554595, ...];
%e A145348 A^8: [1, 8, (60), 520, 5366, 64816, 892308, ...];
%e A145348 A^10: [1, 10, 85, (780), 8190, 98702, 1344920, ...];
%e A145348 A^12: [1, 12, 114, 1108, (11895), 143676, 1943488, ...];
%e A145348 A^14: [1, 14, 147, 1512, 16653, (202384), 2725541, ...]; ...
%e A145348 in which the main diagonal forms the initial terms of this sequence:
%e A145348 [2/2*(1), 2/3*(6), 2/4*(60), 2/5*(780), 2/6*(11895), 2/7*(202384), ...].
%o A145348 (PARI) {a(n)=local(F=1+x,G);for(i=0,n,G=serreverse(x/(F+x*O(x^n))^2)/x;F=1+x*G^2);polcoeff(F,n)}
%o A145348 (PARI) /* This sequence is generated when k=2, m=2: A(x/A(x)^k) = 1 + x*A(x)^m */
%o A145348 {a(n, k=2, m=2)=local(A=sum(i=0, n-1, a(i, k, m)*x^i)+x*O(x^n)); if(n==0, 1, polcoeff((m+k)/(m+k*n)*A^(m+k*n), n-1))}
%o A145348 for(n=0,20,print1(a(n),", "))
%Y A145348 Cf. A145350, A147664, A120972.
%K A145348 nonn
%O A145348 0,3
%A A145348 _Paul D. Hanna_, Nov 09 2008