A385642 G.f. A(x) satisfies 2*(1-x) = Sum_{n=-oo..+oo} (x - A(x)^n)^(n+1) * (A(x) - x^n)^(n+1).

1, 1, 1, 2, 4, 8, 13, 35, 86, 191, 447, 1103, 2810, 6974, 17471, 44795, 115279, 296474, 763834, 1981967, 5164628, 13473784, 35236723, 92443470, 243157407, 640688394, 1691077318, 4472493065, 11849608512, 31441695581, 83545685025, 222309673546, 592337513731, 1580160709355, 4220133780310

Offset

1,4

Comments

a(n)^(1/n) tends to 2.778154... - Vaclav Kotesovec, Jul 06 2025

Links

Paul D. Hanna, Table of n, a(n) for n = 1..401

Formula

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

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

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

Example

G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 4*x^5 + 8*x^6 + 13*x^7 + 35*x^8 + 86*x^9 + 191*x^10 + 447*x^11 + 1103*x^12 + ...

Prog

(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoef(-2*(1-x) + sum(m=-#A, #A, (x - Ser(A)^m)^(m+1) * (Ser(A) - x^m)^(m+1) ), #A-1); ); A[n+1]}
for(n=1, 40, print1(a(n), ", "))

Crossrefs

Sequence in context: A018285 A026665 A369704 * A174540 A354687 A253766

Adjacent sequences: A385639 A385640 A385641 * A385643 A385644 A385645

Keyword

nonn

Author

Paul D. Hanna, Jul 05 2025

Status

approved