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A264407 E.g.f. satisfies: A(x) = 1/(1-x)^(A(x)^2). 1
1, 1, 6, 66, 1084, 23920, 665388, 22374884, 883177328, 40043323728, 2051202965280, 117166763184768, 7384596609153696, 509084508866799840, 38108295339435463296, 3078340850588419228800, 266906341797637061659392, 24724454378396015985551616, 2436960508983873399401081856, 254658073346711773211982974976, 28122779871625104764662272952320 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..20.

FORMULA

E.g.f. also satisfies:

(1) A(x) = sqrt( LambertW(2*log(1-x)) / (2*log(1-x)) ).

(2) A(x) = Sum_{n>=0} (2*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)^2 + k).

(4) A(x) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} |Stirling1(n, k)| * A(x)^(2*k).

a(n) = Sum_{k=0..n} |Stirling1(n, k)| * (2*k+1)^(k-1).

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

EXAMPLE

E.g.f.: A(x) = 1 + x + 6*x^2/2! + 66*x^3/3! + 1084*x^4/4! + 23920*x^5/5! + 665388*x^6/6! + 22374884*x^7/7! + 883177328*x^8/8! +...

where A(x) = 1/(1-x)^(A(x)^2).

From a LambertW identity,

A(x) = 1 - log(1-x) + 5*log(1-x)^2/2! - 7^2*log(1-x)^3/3! + 9^3*log(1-x)^4/4! - 11^4*log(1-x)^5/5! + 13^5*log(1-x)^6/6! +...

Also,

A(x) = 1 + x*A(x)^2 + x^2*A(x)^2*(A(x)^2+1)/2! + x^3*A(x)^2*(A(x)^2+1)*(A(x)^2+2)/3! + x^4*A(x)^2*(A(x)^2+1)*(A(x)^2+2)*(A(x)^2+3)/4! +...

MATHEMATICA

Table[Sum[Abs[StirlingS1[n, k]] * (2*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^2 + k) +x*O(x^n)) ); n!*polcoeff(A, n)}

for(n=0, 20, print1(a(n), ", "))

(PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}

{a(n) = sum(k=0, n, abs(Stirling1(n, k))*(2*k+1)^(k-1))}

for(n=0, 20, print1(a(n), ", "))

CROSSREFS

Cf. A052813, A264408.

Sequence in context: A174496 A008548 A090358 * A112942 A113390 A267080

Adjacent sequences:  A264404 A264405 A264406 * A264408 A264409 A264410

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 17 2015

STATUS

approved

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Last modified April 11 05:05 EDT 2021. Contains 342886 sequences. (Running on oeis4.)