A264408 E.g.f. satisfies: A(x) = 1/(1-x)^(A(x)^3).

1, 1, 8, 123, 2880, 91380, 3670422, 178637046, 10220188584, 672297233904, 49998246211440, 4148400045354048, 379922492043352728, 38069325641907527760, 4143052833221549219112, 486645149764644576491160, 61364015736691449905270208, 8267876533375590035644494336, 1185420892076469924434688605760

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..331

Formula

E.g.f. also satisfies:

(1) A(x) = ( LambertW(3*log(1-x)) / (3*log(1-x)) )^(1/3).

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

(3) A(x) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (A(x)^3 + k).

(4) A(x) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} |Stirling1(n, k)| * A(x)^(3*k).

a(n) = Sum_{k=0..n} |Stirling1(n, k)| * (3*k+1)^(k-1).

a(n) ~ n^(n-1) / (sqrt(3) * exp(n - n*exp(-1)/3 - 5/6) * (exp(exp(-1)/3) - 1)^(n-1/2)). - Vaclav Kotesovec, Nov 18 2015

Example

E.g.f.: A(x) = 1 + x + 8*x^2/2! + 123*x^3/3! + 2880*x^4/4! + 91380*x^5/5! + 3670422*x^6/6! + 178637046*x^7/7! + 10220188584*x^8/8! +...
where A(x) = 1/(1-x)^(A(x)^3).
From a LambertW identity,
A(x) = 1 - log(1-x) + 7*log(1-x)^2/2! - 10^2*log(1-x)^3/3! + 13^3*log(1-x)^4/4! - 16^4*log(1-x)^5/5! + 19^5*log(1-x)^6/6! +...
Also,
A(x) = 1 + x*A(x)^3 + x^2*A(x)^3*(A(x)^3+1)/2! + x^3*A(x)^3*(A(x)^3+1)*(A(x)^3+2)/3! + x^4*A(x)^3*(A(x)^3+1)*(A(x)^3+2)*(A(x)^3+3)/4! +...

Mathematica

Table[Sum[Abs[StirlingS1[n, k]] * (3*k+1)^(k-1), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 18 2015 *)

Prog

(PARI) {a(n) = my(A=1+x); for(i=1, n, A = sum(m=0, n, x^m/m! * prod(k=0, m-1, A^3 + k) +x*O(x^n)) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) a(n) = sum(k=0, n, abs(stirling(n, k, 1))*(3*k+1)^(k-1));
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A052813, A264407, A349505.

Sequence in context: A240477 A195247 A239755 * A364985 A120957 A302356

Adjacent sequences: A264405 A264406 A264407 * A264409 A264410 A264411

Keyword

nonn

Author

Paul D. Hanna, Nov 18 2015

Status

approved