A378574 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^(3*n) * (A(x)^n - 1)^(2*n).

1, 3, 12, 59, 327, 1946, 12134, 78226, 517184, 3487505, 23893646, 165850488, 1163828396, 8242924994, 58847504503, 423033663392, 3059510702675, 22246091978206, 162526826216002, 1192478814606035, 8783088349526872, 64916842264368857, 481332209801339757, 3579243987499087418, 26686563090217529433, 199460739112873554700

Offset

1,2

Comments

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

Links

Paul D. Hanna, Table of n, a(n) for n = 1..500

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)^(3*n) * (A(x)^n - 1)^(2*n).

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

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

a(n) ~ c * d^n / n^(3/2), where d = 7.93146010756761430030130000122704160387300359395... and c = 0.10796855069568137861757704875316294406044834... - Vaclav Kotesovec, Jan 22 2025

Example

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 + ...
where 1/x = Sum_{n=-oo..+oo} A(x)^(3*n) * (A(x)^n - 1)^(2*n).
SPECIFIC VALUES.
A(t) = 1/3 at t = 0.12585895918695186605930233846254446263989590785304...
  where 1/t = Sum_{n=-oo..+oo} (3^n - 1)^(2*n) / 3^(n*(2*n+3)).
A(t) = 1/4 at t = 0.12193627294168244082888935185287294571181842062368...
  where 1/t = Sum_{n=-oo..+oo} (4^n - 1)^(2*n) / 4^(n*(2*n+3)).
A(t) = 1/5 at t = 0.11280607992519488602761908095440027272522717191303...
  where 1/t = Sum_{n=-oo..+oo} (5^n - 1)^(2*n) / 5^(n*(2*n+3)).
A(t) = 1/6 at t = 0.10336638030773187970870206273520530665696515018547...
  where 1/t = Sum_{n=-oo..+oo} (6^n - 1)^(2*n) / 6^(n*(2*n+3)).
A(1/8) = 0.28233353658801389003553692176370867250288852547602...
  where 8 = Sum_{n=-oo..+oo} A(1/8)^(3*n) * (A(1/8)^n - 1)^(2*n).
A(1/9) = 0.19319619869196678103607524467778080039615849113205...
A(1/10) = 0.15679778022503116048268631413823334460078782032231...
A(1/12) = 0.11667708038287135433388818882878921582733641706026...

Prog

(PARI) N=40 \\ number of terms
{a(n) = my(R = sum(m=-N-1, N+1, x^(3*m) * (1 - x^m +x^2*O(x^N))^(2*m) ), A);
A = serreverse(1/R); polcoef(A, n)}
for(n=1, N, print1(a(n), ", "))

Crossrefs

Sequence in context: A181328 A058861 A105668 * A192768 A179325 A064856

Adjacent sequences: A378571 A378572 A378573 * A378575 A378576 A378577

Keyword

nonn

Author

Paul D. Hanna, Jan 10 2025

Status

approved