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A227176 E.g.f.: LambertW(LambertW(-x)) / LambertW(-x). 5
1, 1, 5, 43, 525, 8321, 162463, 3774513, 101808185, 3129525793, 108063152091, 4143297446729, 174723134310277, 8039591465487297, 400924930695585143, 21543513647508536161, 1241094846565489688817, 76314967969651411780673, 4989260143610128556354611 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..100

FORMULA

a(n) = Sum_{k=0..n} binomial(n,k) * k*(k+1)^(k-1) * n^(n-k-1) for n>0 with a(0)=1.

E.g.f. A(x) satisfies:

(1) A(x) = W(x*W(x)), where W(x) = LambertW(-x)/(-x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.

(2) A(x) = exp( T(T(x)) ), where T(x) = -LambertW(-x) is Euler's tree function (A000169).

(3) A(x) = exp( -A(x)*LambertW(-x) ).

(4) A(x) = ( LambertW(-x)/(-x) )^A(x).

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

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

(7) A(x)^m = Sum_{n>=0} m*A(x)*(n + m*A(x))^(n-1) * x^n/n!.

(8) A(x/exp(x)) = exp(T(x)) = LambertW(-x)/(-x).

(9) log(A(x)) = A(x) * Sum_{n>=1} n^(n-1) * x^n/n!, and equals the e.g.f. of A207833.

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

a(n) ~ n! * (-exp((1+exp(-1))*n)/(sqrt(2*Pi*(1-exp(-1)))*n^(3/2) *LambertW(-exp(-1-exp(-1))))). - Vaclav Kotesovec, Jul 05 2013

EXAMPLE

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 43*x^3/3! + 525*x^4/4! + 8321*x^5/5! +...

Define W(x) = LambertW(-x)/(-x), where W(x) = exp(x*W(x)) and begins:

W(x) = 1 + x + 3*x^2/2! + 4^2*x^3/3! + 5^3*x^4/4! + 6^4*x^5/5! +...

then

(1) A(x) = W(x*W(x)),

(4) A(x) = W(x)^A(x),

(3) A(x) = exp( x*A(x)*W(x) ),

(8) A(x/exp(x)) = W(x).

The e.g.f. also satisfies:

(6) A(x) = 1 + A(x)*x + A(x)*(2 + A(x))*x^2/2! + A(x)*(3 + A(x))^2*x^3/3! + A(x)*(4 + A(x))^3*x^4/4! + A(x)*(5 + A(x))^4*x^5/5! +...

and, for all real m,

(7) A(x)^m = 1 + m*A(x)*(1+m*A(x))^0*x^1/1! + m*A(x)*(2+m*A(x))^1*x^2/2! + m*A(x)*(3+m*A(x))^2*x^3/3! + m*A(x)*(4+m*A(x))^3*x^4/4! + m*A(x)*(5+m*A(x))^4*x^5/5! +...

MATHEMATICA

CoefficientList[Series[LambertW[LambertW[-x]]/LambertW[-x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 05 2013 *)

PROG

(PARI) {a(n) = if(n==0, 1, sum(k=0, n, binomial(n, k)*k*(k+1)^(k-1)*n^(n-k-1)))}

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

(PARI) /* E.g.f.: A(x) = W(x*W(x)) */

{a(n)=local(W=sum(k=0, n, (k+1)^(k-1)*x^k/k!)+x*O(x^n)); n!*polcoeff(subst(W, x, x*W), n)}

(PARI) /* E.g.f.: A(x) = exp(T(T(x)) ) */

{a(n)=local(T=sum(k=1, n, k^(k-1)*x^k/k!)+x*O(x^n)); n!*polcoeff(exp(subst(T, x, T)), n)}

(PARI) /* E.g.f.: A(x) = exp( -A(x)*LambertW(-x) ) */

{a(n)=local(A=1+x, LambertW=sum(k=1, n, -k^(k-1)*(-x)^k/k!)+x*O(x^n));

for(i=1, n, A=exp(-A*subst(LambertW, x, -x) +x*O(x^n))); n!*polcoeff(A, n)}

(PARI) /* E.g.f.: A(x) = ( LambertW(-x)/(-x) )^A(x) */

{a(n)=local(A=1+x, W=sum(k=0, n, (k+1)^(k-1)*x^k/k!)+x*O(x^n));

for(i=1, n, A=W^A); n!*polcoeff(A, n)}

(PARI) /* E.g.f.: A(x) = Sum_{n>=0} A(x)*(n + A(x))^(n-1) * x^n/n!. */

{a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, A*(k+A)^(k-1)*x^k/k!)+x*O(x^n)); n!*polcoeff(A, n)}

CROSSREFS

Cf. A207833, A195203, A000169.

Sequence in context: A107720 A299425 A060053 * A132691 A256033 A251568

Adjacent sequences:  A227173 A227174 A227175 * A227177 A227178 A227179

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 04 2013

STATUS

approved

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Last modified December 4 00:33 EST 2020. Contains 338920 sequences. (Running on oeis4.)