A357399 Coefficients of x^n, n >= 0, in A(x) such that: x = Sum_{n=-oo..+oo} (-x)^n * (1 - (-x)^n)^n * A(x)^n.

1, 1, 3, 10, 37, 143, 564, 2270, 9305, 38755, 163569, 698186, 3009129, 13077850, 57250728, 252221229, 1117409653, 4975095073, 22249463540, 99901607730, 450187852401, 2035353779794, 9229671434155, 41968536407303, 191318458136066, 874179701912764, 4002949886221529

Offset: 0

Author

Paul D. Hanna, Nov 07 2022

Keywords

• nonn

Comments

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

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 143*x^5 + 564*x^6 + 2270*x^7 + 9305*x^8 + 38755*x^9 + 163569*x^10 + 698186*x^11 + 3009129*x^12 + ...
such that A = A(x) satisfies
x = ... + (x*A)^(-2)/(1 - x^(-2))^2 - (x*A)^(-1)/(1 + x^(-1)) + 1 - (x*A)*(1 + x) + (x*A)^2*(1 - x^2)^2 - (x*A)^3*(1 + x^3)^3 + (x*A)^4*(1 - x^4)^4 + ... + (-x*A)^n*(1 - (-x)^n)^n + ...
SPECIFIC VALUES.
A(x) = 2 at x = 0.205304925829036018515173714762264122490105588782437881...
A(1/5) = 1.772537703863275044370749593488314266794544517965356320...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A357791.

Programs

PARI

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

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

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