login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A222078 O.g.f.: Sum_{n>=0} n^n*(n+4)^n * exp(-n*(n+4)*x) * x^n / n!. 4
1, 5, 47, 742, 16357, 459369, 15651260, 626935936, 28872594389, 1503262704775, 87328047029511, 5600639046765690, 393092088068927860, 29974039720132943036, 2467669218502361588472, 218168186315818183909344, 20617367868151866462395205, 2074120178028300166990286691 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..340

FORMULA

a(n) = 1/n! * [x^n] Sum_{k>=0} k^k*(k+4)^k * x^k / (1 + k*(k+4)*x)^(k+1).

a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (k+4)^n.

a(n) ~ n^n * 2^(2*n+3/2) / (sqrt(Pi*(1-c)*n) * exp(n) * (2-c)^n * c^(n+2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 22 2014

EXAMPLE

O.g.f.: A(x) = 1 + 5*x + 47*x^2 + 742*x^3 + 16357*x^4 + 459369*x^5 +...

where

A(x) = 1 + 5*x*exp(-5*x) + 12^2*exp(-12*x)*x^2/2! + 21^3*exp(-21*x)*x^3/3! + 32^4*exp(-32*x)*x^4/4! + 45^5*exp(-45*x)*x^5/5! +...

is a power series in x with integer coefficients.

MATHEMATICA

Flatten[{1, Table[Sum[Binomial[n, j] * 4^(n-j) * StirlingS2[n+j, n], {j, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, May 22 2014 *)

PROG

(PARI) {a(n)=polcoeff(sum(m=0, n, m^m*(m+4)^m*x^m*exp(-m*(m+4)*x+x*O(x^n))/m!), n)}

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

(PARI) {a(n)=(1/n!)*polcoeff(sum(k=0, n, k^k*(k+4)^k*x^k/(1+k*(k+4)*x +x*O(x^n))^(k+1)), n)}

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

(PARI) {a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*k^n*(k+4)^n)}

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

CROSSREFS

Cf. A217903, A217900, A222076, A222077, A222079.

Sequence in context: A246040 A183773 A247982 * A061572 A140422 A145564

Adjacent sequences:  A222075 A222076 A222077 * A222079 A222080 A222081

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 24 2013

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 9 16:50 EDT 2020. Contains 335545 sequences. (Running on oeis4.)