A378574 - OEIS

%I #9 Jan 22 2025 04:22:19

%S 1,3,12,59,327,1946,12134,78226,517184,3487505,23893646,165850488,

%T 1163828396,8242924994,58847504503,423033663392,3059510702675,

%U 22246091978206,162526826216002,1192478814606035,8783088349526872,64916842264368857,481332209801339757,3579243987499087418,26686563090217529433,199460739112873554700

%N G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^(3*n) * (A(x)^n - 1)^(2*n).

%C Related identity: Sum_{n=-oo..+oo} x^(3*n) * (1 - x^n)^(n+2) = 0.

%H Paul D. Hanna, <a href="/A378574/b378574.txt">Table of n, a(n) for n = 1..500</a>

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

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

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

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

%F a(n) ~ c * d^n / n^(3/2), where d = 7.93146010756761430030130000122704160387300359395... and c = 0.10796855069568137861757704875316294406044834... - _Vaclav Kotesovec_, Jan 22 2025

%e G.f.: A(x) = x + 3*x^2 + 12*x^3 + 59*x^4 + 327*x^5 + 1946*x^6 + 12134*x^7 + 78226*x^8 + 517184*x^9 + 3487505*x^10 + ...

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

%e SPECIFIC VALUES.

%e A(t) = 1/3 at t = 0.12585895918695186605930233846254446263989590785304...

%e   where 1/t = Sum_{n=-oo..+oo} (3^n - 1)^(2*n) / 3^(n*(2*n+3)).

%e A(t) = 1/4 at t = 0.12193627294168244082888935185287294571181842062368...

%e   where 1/t = Sum_{n=-oo..+oo} (4^n - 1)^(2*n) / 4^(n*(2*n+3)).

%e A(t) = 1/5 at t = 0.11280607992519488602761908095440027272522717191303...

%e   where 1/t = Sum_{n=-oo..+oo} (5^n - 1)^(2*n) / 5^(n*(2*n+3)).

%e A(t) = 1/6 at t = 0.10336638030773187970870206273520530665696515018547...

%e   where 1/t = Sum_{n=-oo..+oo} (6^n - 1)^(2*n) / 6^(n*(2*n+3)).

%e A(1/8) = 0.28233353658801389003553692176370867250288852547602...

%e   where 8 = Sum_{n=-oo..+oo} A(1/8)^(3*n) * (A(1/8)^n - 1)^(2*n).

%e A(1/9) = 0.19319619869196678103607524467778080039615849113205...

%e A(1/10) = 0.15679778022503116048268631413823334460078782032231...

%e A(1/12) = 0.11667708038287135433388818882878921582733641706026...

%o (PARI) N=40 \\ number of terms

%o {a(n) = my(R = sum(m=-N-1, N+1, x^(3*m) * (1 - x^m +x^2*O(x^N))^(2*m) ), A);

%o A = serreverse(1/R); polcoef(A, n)}

%o for(n=1, N, print1(a(n), ", "))

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jan 10 2025