A382323 G.f. A(x) satisfies -3 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1) * A(x)^n.

1, 3, 18, 150, 1323, 12486, 123069, 1253595, 13089576, 139367370, 1507353966, 16515098985, 182913374493, 2044565139303, 23035036108755, 261312501113193, 2982280058702499, 34217698991867058, 394470188685557271, 4566935001939261414, 53076293916648500439, 618991948535588040078

Offset: 0

Author

Paul D. Hanna, Mar 21 2025

Keywords

• nonn

Comments

Related identity: 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1).

Formulas (5) and (6) hold true for all nonzero A(x).

Examples

			G.f.: A(x) = 1 + 3*x + 18*x^2 + 150*x^3 + 1323*x^4 + 12486*x^5 + 123069*x^6 + 1253595*x^7 + 13089576*x^8 + 139367370*x^9 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..301

Crossrefs

Cf. A356783, A380557, A382322.

Programs

PARI

{a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoef(3 + sum(m=-#A, #A, (-1)^m * x^(2*m+1) * (1 + x^m +x*O(x^#A))^(m+1) * Ser(A)^m  ), #A-2); ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following formulas.

(1) -3 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1) * A(x)^n.

(2) 3*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (A(x) + x^n)^(n+1) / A(x)^n.

(3) -3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)+1) / ( (1 + x^n)^(n-1) * A(x)^n ).

(4) 3*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)+1) * A(x)^n / (1 + x^n*A(x))^(n-1).

(5) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n*A(x))^(n+1) / A(x)^n.

(6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)+1) * A(x)^n / (A(x) + x^n)^(n-1).