A380067 G.f. A(x) satisfies 0 = Sum_{n=-oo..+oo} (-x)^n * (A(x) - (-x)^n)^(3*n+1).

1, 2, 9, 76, 605, 5351, 49789, 480401, 4766086, 48292018, 497690157, 5200558352, 54971657745, 586748531155, 6315126497084, 68461134239364, 746869322310992, 8193320926852194, 90327637946283912, 1000230167386765676, 11120012336000921992, 124071390413266176706, 1388854343860145268801

Offset: 0

Author

Paul D. Hanna, Jan 23 2025

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 2*x + 9*x^2 + 76*x^3 + 605*x^4 + 5351*x^5 + 49789*x^6 + 480401*x^7 + 4766086*x^8 + 48292018*x^9 + 497690157*x^10 + ...
SPECIFIC VALUES.
A(t) = 3/2 at t = 0.083194510328352326489147821869211901063348034386178...
A(t) = 4/3 at t = 0.077330132914732168322332648422644564090957191059466...
A(t) = 5/4 at t = 0.069536731258681190910149841234109935263075324366012...
A(t) = 6/5 at t = 0.062338368698372988520462446903734726028227564937896...
A(1/12) = 1.5173629784598497189372653851252639767246421375516...
A(1/13) = 1.3275206734003372155231166649536738395898242302762...
A(1/14) = 1.2663025820443309311247755018250914700758939519601...
A(1/15) = 1.2281465547940615875867824117423734976415492745171...
A(1/16) = 1.2009676782585721612406130744065631283860104847323...
		

Links

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

Crossrefs

Cf. A380068.

Programs

PARI

{a(n) = my(V=[1]); for(i=1,n, V = concat(V,0); A = Ser(V);
V[#V] = -polcoef( sum(n=-#V,#V, (-x)^n * (A - (-x)^n)^(3*n+1) ),#V-1) ); H=A; V[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) 0 = Sum_{n=-oo..+oo} (-x)^n * (A(x) - (-x)^n)^(3*n+1).

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