A358937 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (x^(2*n) - A(x))^n.

1, 1, 3, 6, 13, 31, 76, 192, 504, 1351, 3668, 10082, 27991, 78335, 220778, 626141, 1785593, 5117179, 14729826, 42568767, 123465517, 359268141, 1048541699, 3068583485, 9002849260, 26474484680, 78019959584, 230381635121, 681544367457, 2019718168994, 5995000501189

Offset: 0

Author

Paul D. Hanna, Dec 07 2022

Keywords

• nonn

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^(2*n+1))^n, which holds formally for all y.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 13*x^4 + 31*x^5 + 76*x^6 + 192*x^7 + 504*x^8 + 1351*x^9 + 3668*x^10 + 10082*x^11 + 27991*x^12 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A366229, A355865.

Programs

PARI

{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff( sum(n=-#A,#A, x^n * (x^(2*n) - Ser(A))^n ), #A) );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) 1 = Sum_{n=-oo..+oo} x^n * (x^(2*n) - A(x))^n.

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

(3) 0 = Sum_{n=-oo..+oo} (-1)^n * x^n * (x^(2*n+1) - A(x))^n (trivial).

(4) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n*(n-1)) / (1 - x^(2*n)*A(x))^n (trivial).