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 A363556 #16 Jul 20 2023 10:07:00
%S A363556 1,1,4,20,116,729,4835,33308,236040,1709710,12601576,94209348,
%T A363556 712665643,5445033052,41957720686,325702058100,2544612673624,
%U A363556 19993230400674,157880283331190,1252345341860312,9974116387535508,79727867712667593,639419309923767255,5143690907082615272
%N A363556 Expansion of g.f. A(x) satisfying 0 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * A(x)^(n^2) / Product_{k=2..n+2} (1 - x^k*A(x)^(2*k+1)).
%C A363556 Related identities:
%C A363556 (1) 0 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * D(x)^(n^2) / Product_{k=1..n+1} (1 - x^k*D(x)^(2*k+1)), where D(x) = 1 + x*D(x)^3 is the g.f. of A001764.
%C A363556 (2) 1 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * F(x)^(n^2) / Product_{k=0..n} (1 - x^k*F(x)^(2*k+1)), which holds as a formal power series for all x and F(x).
%C A363556 (3) Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * y^(n^2) / Product_{k=2..n+2} (1 - x^k*y^(2*k+1)) = (1 - y) + y^4*x + (y^5 - y^6)*x^2 - y^8*x^3 + y^10*x^4.
%H A363556 Paul D. Hanna, <a href="/A363556/b363556.txt">Table of n, a(n) for n = 0..400</a>
%F A363556 G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
%F A363556 (1) A(x) = 1 + A(x)^4*x + (A(x)^5 - A(x)^6)*x^2 - A(x)^8*x^3 + A(x)^10*x^4.
%F A363556 (2) A(x) = F(x*A(x)) where F(x) = A(x/F(x)) is the g.f. of A363555.
%F A363556 (3) A(x) = G(x*A(x)^2) where G(x) = A(x/G(x)^2) = (1 + x - sqrt(1 - 2*x - 3*x^2))/(2*x*(1-x^2)) is the g.f. of A082395 (with an offset of 0).
%F A363556 (4) 0 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * A(x)^(n^2) / Product_{k=2..n+2} (1 - x^k*A(x)^(2*k+1)).
%e A363556 G.f.: A(x) = 1 + x + 4*x^2 + 20*x^3 + 116*x^4 + 729*x^5 + 4835*x^6 + 33308*x^7 + 236040*x^8 + 1709710*x^9 + 12601576*x^10 + ...
%e A363556 such that A = A(x) satisfies
%e A363556 0 = 1/(1-x^2*A^5) - A/((1-x^2*A^5)*(1-x^3*A^7)) + x*A^4/((1-x^2*A^5)*(1-x^3*A^7)*(1-x^4*A^9)) - x^3*A^9/((1-x^2*A^5)*(1-x^3*A^7)*(1-x^4*A^9)*(1-x^5*A^11)) + x^6*A^16/((1-x^2*A^5)*(1-x^3*A^7)*(1-x^4*A^9)*(1-x^5*A^11)*(1-x^6*A^13)) -+ ...
%e A363556 Compare to an identity of the function D = 1 + x*D^3 (A001764), where
%e A363556 0 = 1/(1-x*D^3) - D/((1-x*D^3)*(1-x^2*D^5)) + x*D^4/((1-x*D^3)*(1-x^2*D^5)*(1-x^3*D^7)) - x^3*D^9/((1-x*D^3)*(1-x^2*D^5)*(1-x^3*D^7)*(1-x^4*D^9)) + x^6*D^16/((1-x*D^3)*(1-x^2*D^5)*(1-x^3*D^7)*(1-x^4*D^9)*(1-x^5*D^11)) -+ ...
%o A363556 (PARI) /* As a solution to a polynomial equation */
%o A363556 {a(n) = my(A=1); for(i=1, n, A = 1 + A^4*x + (A^5 - A^6)*x^2 - A^8*x^3 + A^10*x^4 +x*O(x^n)); polcoeff(A, n)}
%o A363556 for(n=0, 30, print1(a(n), ", "))
%o A363556 (PARI) /* By Series Reversion involving a g.f. for A082395 */
%o A363556 {a(n) = my(A=1, G082395 = (1 + x - sqrt(1 - 2*x - 3*x^2 +O(x^(n+2))))/(2*x*(1-x^2)) ); A = sqrt( (1/x)*serreverse( x/G082395^2 ) ); polcoeff(A, n)}
%o A363556 for(n=0, 30, print1(a(n), ", "))
%o A363556 (PARI) /* By Definition */
%o A363556 {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A363556 A[#A] = polcoeff( sum(m=0,2*sqrtint(#A), (-1)^m * (x)^(m*(m-1)/2) * Ser(A)^(m^2) / prod(k=2,m+2, (1 - 1*x^k*Ser(A)^(2*k+1) ) )),#A-1);); H=A; A[n+1]}
%o A363556 for(n=0,30,print1(a(n),", "))
%Y A363556 Cf. A082395, A363555.
%K A363556 nonn
%O A363556 0,3
%A A363556 _Paul D. Hanna_, Jul 09 2023