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A122399 a(n) = Sum_{k=0..n} k^n * k! * Stirling2(n,k). 22
1, 1, 9, 211, 9285, 658171, 68504709, 9837380491, 1863598406805, 450247033371451, 135111441590583909, 49300373690091496171, 21495577955682021043125, 11037123350952586270549531, 6591700149366720366704735109 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

FORMULA

E.g.f.: Sum((exp(n*x) - 1)^n, n=0..infinity). - Vladeta Jovovic, Sep 03 2006

E.g.f.: Sum_{n>=0} exp(n^2*x) / (1 + exp(n*x))^(n+1). - Paul D. Hanna, Oct 26 2014

E.g.f.: Sum_{n>=0} exp(-n*x) / (1 + exp(-n*x))^(n+1). - Paul D. Hanna, Oct 30 2014

O.g.f.: Sum_{n>=0} n^n * n! * x^n / Product_{k=1..n} (1 - n*k*x). - Paul D. Hanna, Jan 05 2013

Limit n->infinity (a(n)/n!)^(1/n)/n = ((1+exp(1/r))*r^2)/exp(1) = A317855/exp(1) = 1.162899527477400818845..., where r = 0.87370243323966833... is the root of the equation 1/(1+exp(-1/r)) = -r*LambertW(-exp(-1/r)/r). - Vaclav Kotesovec, Jun 21 2013

a(n) ~ c * A317855^n * (n!)^2 / sqrt(n), where c = 0.327628285569869481442286492410507030710253054522608... - Vaclav Kotesovec, Aug 09 2018

EXAMPLE

E.g.f.: A(x) = 1 + x + 9*x^2/2! + 211*x^3/3! + 9285*x^4/4! + 658171*x^5/5! +...

such that

A(x) = 1 + (exp(x)-1) + (exp(2*x)-1)^2 + (exp(3*x)-1)^3 + (exp(4*x)-1)^4 +...

The e.g.f. is also given by the series:

A(x) = 1/2 + exp(x)/(1+exp(x))^2 + exp(4*x)/(1+exp(2*x))^3 + exp(9*x)/(1+exp(3*x))^4 + exp(16*x)/(1+exp(4*x))^5 + exp(25*x)/(1+exp(5*x))^6 +...

or, equivalently,

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

MAPLE

a := n -> add(k^n*k!*combinat[stirling2](n, k), k=0..n); # Max Alekseyev, Feb 01 2007

MATHEMATICA

Flatten[{1, Table[Sum[k^n*k!*StirlingS2[n, k], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jun 21 2013 *)

PROG

(PARI) {a(n)=polcoeff(sum(m=0, n, m^m*m!*x^m/prod(k=1, m, 1-m*k*x+x*O(x^n))), n)}

for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 05 2013

(PARI) {a(n)=n!*polcoeff(sum(k=0, n, (exp(k*x +x*O(x^n)) - 1)^k), n)}

for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Oct 26 2014

(PARI) /* From e.g.f. infinite series: */

\p100 \\ set precision

{A=Vec(serlaplace(sum(n=0, 500, 1.*exp(n^2*x +O(x^26))/(1 + exp(n*x +O(x^26)))^(n+1)) ))}

for(n=0, #A-1, print1(round(A[n+1]), ", ")) \\ Paul D. Hanna, Oct 30 2014

CROSSREFS

Cf. A122400, A220181, A320083.

Sequence in context: A001535 A300136 A274672 * A320096 A188409 A109587

Adjacent sequences:  A122396 A122397 A122398 * A122400 A122401 A122402

KEYWORD

easy,nonn

AUTHOR

Vladeta Jovovic, Aug 31 2006

EXTENSIONS

More terms from Max Alekseyev, Feb 01 2007

STATUS

approved

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Last modified April 25 12:04 EDT 2019. Contains 322456 sequences. (Running on oeis4.)