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.

1, 0, 0, 2, 0, 0, 8, 0, 0, 44, 0, 6, 280, 0, 96, 1934, 0, 1124, 14088, 18, 11792, 106536, 648, 117626, 828360, 13416, 1142288, 6580780, 216000, 10921088, 53184864, 3019614, 103408416, 435930008, 38629656, 973041448, 3615741192, 465419760, 9118011128, 30298375236

Offset: 1

Author

Paul D. Hanna, Jun 02 2022

Keywords

• nonn

Examples

			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 + ...
such that A = A(x) satisfies:
(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 +--+ ...
(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 + ...
(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 + ...
(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) * ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..400
• Vaclav Kotesovec, Plot of a(n+1)/a(n) for n = 18..400

Crossrefs

Cf. A354649, A354650, A354662, A354663, A354664, A268299, A354652, A354653, A354654.

Programs

PARI

{a(n) = my(A=[0]); for(i=0,n, A = concat(A,0);
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]}
for(n=1,50,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:

(1) 1 = Sum_{n=-oo..oo} (-x)^(n*(n-1)/2) * A(x)^(n*(n+1)/2).

(2) 1 = Sum_{n>=0} (-x)^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x)^(n*(n+1)/2).

(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).

(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.

(5) A(-A(-x)) = x.

a(n) = (-1)^n * Sum_{k=0..2*n+1} A354649(n,k), for n >= 0.

a(n) = -Sum_{k=0..2*n+1} A354650(n,k)*(-1)^k, for n >= 0.