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A179929 a(n) = 2^n*A(n, -1/2), A(n, x) the Eulerian polynomials. 3

%I #31 Jun 26 2019 10:05:55

%S 1,2,2,-6,-30,42,882,954,-39870,-203958,2300562,29677914,-120958110,

%T -4657703958,-7059175758,807984602874,6667870853250,-145556787011958,

%U -2827006784652078,21703953751815834,1108558810703202210

%N a(n) = 2^n*A(n, -1/2), A(n, x) the Eulerian polynomials.

%H Stanislav Sykora, <a href="/A179929/b179929.txt">Table of n, a(n) for n = 0..199</a>

%H OEIS Wiki, <a href="http://oeis.org/wiki/Eulerian_polynomials"> Eulerian polynomials</a>

%F From _Philippe Deléham_, Dec 22 2011: (Start)

%F a(n) = Sum_{k, 0<=k<=n} A123125(n,k)*(-1)^(n-k)*2^k

%F a(n) = Sum_{k, 0<=k<=n} A173018(n,k)*2^(n-k)*(-1)^k. (End)

%F From _Peter Bala_, Mar 12 2013: (Start)

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

%F Recurrence equation: a(n+1) = 3a(n) - Sum_{k=0..n} binomial(n,k) a(k)a(n-k).

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

%F 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

%F From _Stanislav Sykora_, May 15 2014: (Start)

%F a(n) = -2*A212846(n) for n > 0.

%F a(n) = -3^(n+1)*Li(-n, -1/2), with Li(-n, x) = Sum_{k>=0} ((k^n)*(x^k)) the polylogarithm.

%F a(n) = Sum_{k=0..n} 3^(n-k)*(-1)^k*k!*S(n+1, k+1), S(m, l) the Stirling number of second kind. (End)

%t a[n_] := Sum[3^(n-k) (-1)^k k! StirlingS2[n+1, k+1], {k, 0, n}];

%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jun 17 2019 *)

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

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

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

%o return(s);} // - _Stanislav Sykora_, May 15 2014

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

%Y Cf. A212846, A123125, A173018, A195205.

%K sign

%O 0,2

%A _Peter Luschny_, Aug 03 2010

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)