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A124311 a(n) = Sum_{i=0..n} (-2)^i*B(i)*binomial(n,i) where B(n) = Bell numbers A000110(n). 7
1, -1, 5, -21, 121, -793, 5917, -49101, 447153, -4421105, 47062773, -535732805, 6484924585, -83079996041, 1121947980173, -15915567647101, 236442490569825, -3668776058118881, 59316847871113445, -997182232031471477, 17397298225094055897, -314449131128077197561 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The sequence has strictly alternating signs. The variant Dobinski-type formula e^(-1)* (2)^n * sum( (k-1/2)^n / k!,k=0..infinity) is strictly positive. - Karol A. Penson and Olivier Gérard, Oct 22 2007

LINKS

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

FORMULA

E.g.f.: exp(exp(-2*x)-1+x). - Vladeta Jovovic, Aug 04 2007

G.f.: 1/U(0) where U(k)= 1 + x*(2*k+1) - 4*x^2*(k+1)/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 11 2012

a(n) ~ (-2)^n * n^(n - 1/2) * exp(n/LambertW(n) - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n - 1/2)). - Vaclav Kotesovec, Jun 26 2022

MATHEMATICA

Table[ Sum[ (-2)^(k) Binomial[n, k] BellB[k], {k, 0, n}], {n, 0, 50}] (* Karol A. Penson and Olivier Gérard, Oct 22 2007 *)

With[{nn=30}, CoefficientList[Series[Exp[Exp[-2x]-1+x], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Mar 04 2016 *)

PROG

(Sage)

def A124311_list(n): # n>=1

T = [0]*(n+1); R = [1]

for m in (1..n-1):

a, b, c = 1, 0, 0

for k in range(m, -1, -1):

r = a + 2*(k*(b+c)+c)

if k < m : T[k+2] = u;

a, b, c = T[k-1], a, b

u = r

T[1] = u;

R.append((-1)^m*sum(T))

return R

A124311_list(22) # Peter Luschny, Nov 02 2012

CROSSREFS

Cf. A000110, A000296, A005493, A126390, A126617.

Sequence in context: A168598 A002711 A218962 * A353736 A208593 A213009

Adjacent sequences: A124308 A124309 A124310 * A124312 A124313 A124314

KEYWORD

sign

AUTHOR

N. J. A. Sloane, Aug 04 2007

STATUS

approved

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Last modified February 7 22:58 EST 2023. Contains 360132 sequences. (Running on oeis4.)