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A126617 a(n) = Sum_{i=0..n} (-2)^(n-i)*B(i)*binomial(n,i) where B(n) = Bell numbers A000110(n). 7
1, -1, 2, -3, 7, -10, 31, -21, 204, 307, 2811, 12100, 74053, 432211, 2768858, 18473441, 129941283, 956187814, 7351696139, 58897405759, 490681196604, 4242903803727, 38014084430983, 352341755256348, 3373662303816313, 33326335433122711, 339232538387804530 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) is positive starting at n=8. - Karol A. Penson and Olivier Gérard, Oct 22 2007

Hankel transform is A000178. [Paul Barry, Apr 23 2009]

LINKS

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

FORMULA

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

a(n) = e^(-1) * sum( (k-2)^n / k!,k = 0..infinity ). This is a Dobinski-type formula. - Karol A. Penson and Olivier Gérard, Oct 22 2007

G.f.: 1/(1+x-x^2/(1-2x^2/(1-x-3x^2/(1-2x-4x^2/(1-3x-5x^2/(1-.... (continued fraction). [Paul Barry, Apr 23 2009]

Let A be the upper Hessenberg matrix of order n defined by: A[i,i-1]=-1, A[i,j]=binomial(j-1,i-1), (i<=j), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=(-1)^(n)charpoly(A,2). [Milan Janjic, Jul 08 2010]

G.f.: -1/U(0)  where U(k) = x*k - 1 - x - x^2*(k+1)/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Sep 28 2012

G.f.: 1/G(0) where G(k) = 1 + 2*x/(1 + 1/(1 - 2*x*(k+1)/G(k+1)));(continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 23 2012

G.f.: G(0)/(1+3*x) where G(k) = 1 - 2*x*(k+1)/((2*k+1)*(2*x*k-2*x-1) - x*(2*k+1)*(2*k+3)*(2*x*k-2*x-1)/(x*(2*k+3) - 2*(k+1)*(2*x*k-x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 19 2012

From Sergei N. Gladkovskii, Feb 13 2013: (Start)

Conjecture: if the e.g.f. is E(x)= exp( exp(x) -1 + p*x) then

G.f.: (x+1-p*x)/x/(G(0)-x) - 1/x where G(k) = 2*x + 1 - p*x - x*k + x*(x*k - x - 1 + p*x)/G(k+1); (continued fraction).

So, for this sequence (p=-2), g.f.: (3*x+1)/x/( G(0)-x ) - 1/x where G(k) = 4*x + 1 - x*k + x*(x*k - 3*x - 1)/G(k+1);

(End)

G.f.: 1/Q(0), where Q(k) = 1 + 2*x - x/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013

EXAMPLE

G.f.: 1 - 1*x + 2*x^2 - 3*x^3 + 7*x^4 - 10*x^5 + 31*x^6 - 21*x^7 + 204*x^8 + 307*x^9 + 2811*x^10 + 12100*x^11 + 74053*x^12 + 432211*x^13 + ...

MATHEMATICA

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

CROSSREFS

Cf. A000110, A000296, A005493, A124311, A126390, A153732.

Sequence in context: A062113 A130968 A007748 * A117733 A066236 A118203

Adjacent sequences:  A126614 A126615 A126616 * A126618 A126619 A126620

KEYWORD

sign

AUTHOR

N. J. A. Sloane, Aug 04 2007

EXTENSIONS

More terms from Karol A. Penson and Olivier Gérard, Oct 22 2007

STATUS

approved

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Last modified February 21 19:30 EST 2019. Contains 320376 sequences. (Running on oeis4.)