%I #54 Jun 08 2023 08:51:33
%S 1,2,3,2,-3,2,63,2,-1383,2,50523,2,-2702763,2,199360983,2,
%T -19391512143,2,2404879675443,2,-370371188237523,2,69348874393137903,
%U 2,-15514534163557086903,2,4087072509293123892363,2,-1252259641403629865468283,2,441543893249023104553682823
%N Expansion of e.g.f. exp(2x)*sech(x).
%C Transform of 2^n under the matrix A119879.
%C Also the Swiss-Knife polynomials A153641 evaluated at x=2. - _Peter Luschny_, Nov 23 2012
%H Vincenzo Librandi, <a href="/A119880/b119880.txt">Table of n, a(n) for n = 0..200</a>
%H Paul Barry, <a href="https://arxiv.org/abs/1702.04007">Eulerian-Dowling Polynomials as Moments, Using Riordan Arrays</a>, arXiv:1702.04007 [math.CO], 2017.
%F a(n) = Sum_{k=0..n} A119879(n,k) * 2^k.
%F From _Sergei N. Gladkovskii_, Oct 14 2012 to Dec 16 2013: (Start)
%F Continued fractions:
%F G.f.: 1/U(0) where U(k) = 1 - x - x*(k+1)/(1 + x*(k+1)/U(k+1)).
%F G.f.: 1/Q(0), where Q(k) = 1 - 3*x + x*(k+1)/(1-x*(k+1)/Q(k+1)).
%F G.f.: x/(1-x)/Q(0) + 1/(1-x), where Q(k) = 1 - x + x^2*(k+1)*(k+2)/Q(k+1).
%F G.f.: T(0)/(1-2*x), where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2 + (1-2*x)^2/T(k+1)).
%F E.g.f.: 2/Q(0), where Q(k) = 1 + 3^k/( 1 - x/( x - 3^k*(k+1)/Q(k+1))). (End)
%F a(n) = 2*(1+zeta(-n)*(2^n-1)+2^(2*n+1)*zeta(-n,3/4)). - _Peter Luschny_, Jul 16 2013
%F a(n) = (-2)^n*Euler(n, -1/2). - _Peter Luschny_, Jul 21 2020
%p A119880_list := proc(n) local S,A,m,k;
%p A := array(0..n-1,0..n-1); S := NULL;
%p for m from 0 to n-1 do
%p A[m,0] := (-2)^m*euler(m,0);
%p for k from m-1 by -1 to 0 do
%p A[k,m-k] := A[k+1,m-k-1] + A[k,m-k-1] od;
%p S := S,A[0,m] od;
%p S end:
%p A119880_list(31); # _Peter Luschny_, Jun 15 2012
%p P := proc(n,x) option remember; if n = 0 then 1 else
%p (n*x-(1/2)*(1-x))*P(n-1,x)+x*(1-x)*diff(P(n-1,x),x);
%p expand(%) fi end:
%p A119880 := n -> (-1)^n*subs(x=-1, P(n,x)):
%p seq(A119880(n), n=0..30); # _Peter Luschny_, Mar 07 2014
%t Table[2 (1 + Zeta[-n] (2^n - 1) + 2^(2n+1) Zeta[-n, 3/4]), {n, 0, 30}] (* _Peter Luschny_, Jul 16 2013 *)
%t Range[0, 30]! CoefficientList[Series[Exp[2 x] Sech[x], {x, 0, 30}], x] (* _Vincenzo Librandi_, Mar 08 2014 *)
%o (Sage)
%o def skp(n, x):
%o A = lambda k: 0 if (k+1)%4 == 0 else (-1)^((k+1)//4)*2^(-(k//2))
%o return add(A(k)*add((-1)^v*binomial(k,v)*(v+x+1)^n for v in (0..k)) for k in (0..n))
%o A119880 = lambda n: skp(n,2)
%o [A119880(n) for n in (0..30)] # _Peter Luschny_, Nov 23 2012
%o (Magma)
%o EulerPoly:= func< n,x | (&+[ (&+[ (-1)^j*Binomial(k,j)*(x+j)^n : j in [0..k]])/2^k: k in [0..n]]) >;
%o A119880:= func< n | (-2)^n*EulerPoly(n, -1/2) >;
%o [A119880(n): n in [0..40]]; // _G. C. Greubel_, Jun 07 2023
%Y Cf. A119879, A119880, A122045.
%K easy,sign
%O 0,2
%A _Paul Barry_, May 26 2006