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A354661 G.f. A(x) satisfies: 1 = Sum_{n=-oo..oo} (-x)^(n*(n+1)/2) * A(x)^(n*(n-1)/2), with A(0) = 0.

%I A354661 #11 Mar 19 2023 08:06:23
%S A354661 1,0,0,2,0,0,8,0,0,44,0,6,280,0,96,1934,0,1124,14088,18,11792,106536,
%T A354661 648,117626,828360,13416,1142288,6580780,216000,10921088,53184864,
%U A354661 3019614,103408416,435930008,38629656,973041448,3615741192,465419760,9118011128,30298375236
%N A354661 G.f. A(x) satisfies: 1 = Sum_{n=-oo..oo}  (-x)^(n*(n+1)/2) * A(x)^(n*(n-1)/2), with A(0) = 0.
%H A354661 Paul D. Hanna, <a href="/A354661/b354661.txt">Table of n, a(n) for n = 1..400</a>
%H A354661 Vaclav Kotesovec, <a href="/A354661/a354661.jpg">Plot of a(n+1)/a(n) for n = 18..400</a>
%F A354661 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
%F A354661 (1) 1 = Sum_{n=-oo..oo}  (-x)^(n*(n-1)/2) * A(x)^(n*(n+1)/2).
%F A354661 (2) 1 = Sum_{n>=0}  (-x)^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x)^(n*(n+1)/2).
%F A354661 (3) 1 = Sum_{n>=0}  (-1)^(n*(n+1)/2) * A(x)^(n*(n-1)/2) * (1 + A(x)^(2*n+1)) * x^(n*(n+1)/2).
%F A354661 (4) 1 = Product_{n>=1}  (1 - (-x)^n*A(x)^n) * (1 + (-x)^(n-1)*A(x)^n) * (1 + (-x)^n*A(x)^(n-1)), by the Jacobi triple product identity.
%F A354661 (5) A(-A(-x)) = x.
%F A354661 a(n) = (-1)^n * Sum_{k=0..2*n+1} A354649(n,k), for n >= 0.
%F A354661 a(n) = -Sum_{k=0..2*n+1} A354650(n,k)*(-1)^k, for n >= 0.
%e A354661 G.f.: A(x) = x + 2*x^4 + 8*x^7 + 44*x^10 + 6*x^12 + 280*x^13 + 96*x^15 + 1934*x^16 + 1124*x^18 + 14088*x^19 + 18*x^20 + 11792*x^21 + ...
%e A354661 such that A = A(x) satisfies:
%e A354661 (1) 1 = ... + x^36*A^28 + x^28*A^21 - x^21*A^15 - x^15*A^10 + x^10*A^6 + x^6*A^3 - x^3*A - x + 1 + A - x*A^3 - x^3*A^6 + x^6*A^10 + x^10*A^15 - x^15*A^21 - x^21*A^28 + x^28*A^36 +--+ ...
%e A354661 (2) 1 = (1-x) + (1-x^3)*A - x*(1-x^5)*A^3 - x^3*(1-x^7)*A^6 + x^6*(1-x^9)*A^10 + x^10*(1-x^11)*A^15 - x^15*(1-x^13)*A^21 - x^21*(1-x^15)*A^28 + ...
%e A354661 (3) 1 = (1+A) - (1+A^3)*x - A*(1+A^5)*x^3 + A^3*(1+A^7)*x^6 + A^6*(1+A^9)*x^10 - A^10*(1+A^11)*x^15 - A^15*(1+A^13)*x^21 + A^21*(1+A^15)*x^28 + ...
%e A354661 (4) 1 = (1 + x*A)*(1 + A)*(1-x) * (1 - x^2*A^2)*(1 - x*A^2)*(1 + x^2*A) * (1 + x^3*A^3)*(1 + x^2*A^3)*(1 - x^3*A^2) * (1 - x^4*A^4)*(1 - x^3*A^4)*(1 + x^4*A^3) * (1 + x^5*A^5)*(1 + x^4*A^5)*(1 - x^5*A^4) * ...
%o A354661 (PARI) {a(n) = my(A=[0]); for(i=0,n, A = concat(A,0);
%o A354661 A[#A] = -polcoeff(-1 + sum(m=0,sqrtint(2*#A+9), (-x)^(m*(m-1)/2) * (1 - x^(2*m+1)) * Ser(A)^(m*(m+1)/2) ),#A-1) );H=A;A[n+1]}
%o A354661 for(n=1,50,print1(a(n),", "))
%Y A354661 Cf. A354649, A354650, A354662, A354663, A354664, A268299, A354652, A354653, A354654.
%K A354661 nonn
%O A354661 1,4
%A A354661 _Paul D. Hanna_, Jun 02 2022