A357221 Coefficients in the power series A(x) such that: x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.

1, 1, 2, 8, 26, 97, 361, 1399, 5532, 22318, 91387, 379037, 1588769, 6720065, 28645624, 122937300, 530748439, 2303446566, 10043922651, 43979954296, 193309569331, 852599816069, 3772220833468, 16737583785420, 74461239372631, 332062396407641, 1484162266154404

Offset: 0

Author

Paul D. Hanna, Sep 18 2022

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 26*x^4 + 97*x^5 + 361*x^6 + 1399*x^7 + 5532*x^8 + 22318*x^9 + 91387*x^10 + 379037*x^11 + 1588769*x^12 + ...
such that
x*A(x) = ... + x^12/A(x)^4 - x^6/A(x)^3 + x^2/A(x)^2 - 1/A(x) + 1 - x^2*A(x) + x^6*A(x)^2 - x^12*A(x)^3 + x^20*A(x)^4 + ... + (-1)^n * x^(n*(n+1)) * A(x)^n + ...
		

Crossrefs

Cf. A355357, A355361, A357222, A357223, A357224, A357225, A357226.

Programs

PARI

{a(n,p=1) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( x*Ser(A)^p - sum(m=-ceil(sqrt(n+1)), ceil(sqrt(n+1)), (-1)^m*x^(m*(m+1))*Ser(A)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:

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

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

(3) x*A(x) = Product_{n>=1} (1 - x^(2*n)*A(x)) * (1 - x^(2*n-2)/A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.

(4) -x*A(x)^2 = Product_{n>=1} (1 - x^(2*n)/A(x)) * (1 - x^(2*n-2)*A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.