A379202 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 2)^(n+1).

1, 4, 20, 122, 850, 6432, 51324, 424694, 3608592, 31291658, 275774228, 2462835772, 22239367632, 202713590686, 1862689951724, 17235880764264, 160466865121154, 1502055108051124, 14127846520455180, 133455751612975948, 1265563747442829216, 12043611154775588194, 114978748131733714360

Offset: 1

Author

Paul D. Hanna, Dec 20 2024

Keywords

• nonn

Comments

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

Conjecture: a(n) is even for n > 1.

It appears that a(n) == 2 (mod 4) at n = A028309(k) for k >= 4.

Examples

			G.f.: A(x) = x + 4*x^2 + 20*x^3 + 122*x^4 + 850*x^5 + 6432*x^6 + 51324*x^7 + 424694*x^8 + 3608592*x^9 + 31291658*x^10 + ...
SPECIFIC VALUES.
A(t) = 1/6 at t = 0.090270773138940793847220645261976952310511883470512...
  where t = 1/Sum_{n=-oo..+oo} (1 + 2*6^(n-1))^n / 6^(n^2-1).
A(t) = 1/7 at t = 0.084362907984862824662513569761745773472320783010611...
  where t = 1/Sum_{n=-oo..+oo} (1 + 2*7^(n-1))^n / 7^(n^2-1).
A(t) = 1/8 at t = 0.078703999402417120618295617221021413542415048822164...
  where t = 1/Sum_{n=-oo..+oo} (1 + 2*8^(n-1))^n / 8^(n^2-1).
A(1/11) = 0.16976727159020613475135380983780463368461713164010...
A(1/12) = 0.13933682309394427848416123650354034389806333559384...
A(1/15) = 0.09515898887066227963795425335824195002284059150209...
A(1/20) = 0.06369786461564277053938913595571090186089127528505...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..300

Crossrefs

Cf. A379200, A379199, A166952, A378264, A379203, A379204, A379205.

Programs

PARI

{a(n) = my(V=[0, 1], A); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(m=-#A, #A, A^m*(A^m + 2)^(m+1) ), #V-3); ); polcoef(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

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

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

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

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

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

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