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A179929 2^n A(n,-1/2), A(n,x) the Eulerian polynomials. 3
1, 2, 2, -6, -30, 42, 882, 954, -39870, -203958, 2300562, 29677914, -120958110, -4657703958, -7059175758, 807984602874, 6667870853250, -145556787011958, -2827006784652078, 21703953751815834, 1108558810703202210 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Stanislav Sykora, Table of n, a(n) for n = 0..199

OEIS Wiki, Eulerian polynomials

FORMULA

a(n) = Sum_{k, 0<=k<=n} A123125(n,k)*(-1)^(n-k)*2^k = Sum_{k, 0<=k<=n} A173018(n,k)*2^(n-k)*(-1)^k. - Philippe Deléham, Dec 22 2011

From Peter Bala, Mar 12 2013: (Start)

E.g.f.: 3/(1 + 2exp(-3x)) = 1 + 2x + 2x^2/2! - 6x^3/3! - 30x^4/4! + ....

Recurrence equation: a(n+1) = 3a(n) - sum {k = 0..n} binomial(n,k) a(k)a(n-k), with a(0) = 1.

(-1)^n*a(n) are the coefficients of a delta operator associated with a sequence of polynomials of binomial type - see A195205.

(End)

a(n) ~ n! * 2*3^(n+1)/(Pi^2+(log(2))^2)^(n/2+1) * (Pi*sin(n*arctan(Pi/log(2))) - log(2)*cos(n*arctan(Pi/log(2)))). - Vaclav Kotesovec, Oct 09 2013

From Stanislav Sykora, May 15 2014: (Start)

For n>0, a(n) = -2*A212846(n).

a(n) = -3^(n+1)*Li(-n,-1/2), with Li(-n,x)=Sum[k=0..inf]((k^n)*(x^k)) being the polylogarithm.

Also a(n) = Sum[k=0..n](3^(n-k)*(-1)^k*k!*S(n+1,k+1), S(m,l) being the Stirling number of second kind.

(End)

PROG

(PARI) A179929(n) = {local(s, k, term);

  term = 3^n; s = term*stirling(n+1, 1, 2);

  for (k=1, n, term *= -k/3; s += term*stirling(n+1, k+1, 2); );

return(s); } // - Stanislav Sykora, May 15 2014

CROSSREFS

Cf. A000629 = 2^n A(n, 1/2).

Cf. A195205, A212846.

Sequence in context: A108800 A270487 A058250 * A278258 A067644 A184312

Adjacent sequences:  A179926 A179927 A179928 * A179930 A179931 A179932

KEYWORD

sign

AUTHOR

Peter Luschny, Aug 03 2010

STATUS

approved

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Last modified May 23 08:43 EDT 2017. Contains 286909 sequences.