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A119881 Expansion of e.g.f. exp(3*x)*sech(x).

%I A119881 #49 Jun 08 2023 08:51:20
%S A119881 1,3,8,18,32,48,128,528,512,-6912,2048,357888,8192,-22351872,32768,
%T A119881 1903822848,131072,-209865080832,524288,29088886161408,2097152,
%U A119881 -4951498048929792,8388608,1015423886523629568,33554432,-246921480190140874752,134217728,70251601603944228323328
%N A119881 Expansion of e.g.f. exp(3*x)*sech(x).
%C A119881 Transform of 3^n under the matrix A119879.
%C A119881 Also the Swiss-Knife polynomials A153641 evaluated at x=3. - _Peter Luschny_, Nov 23 2012
%H A119881 Vincenzo Librandi, <a href="/A119881/b119881.txt">Table of n, a(n) for n = 0..200</a>
%F A119881 a(n) = Sum_{k=0..n} A119879(n,k)*3^k.
%F A119881 a(n) = Sum_{k=0..n} binomial(n,k)*B(k,1)*2^(n+k)/(n-k+1). Here B(k,1) are the Bernoulli number A027641(k)/A027642(k) with the exception B(1,1)=1/2. - _Peter Luschny_, Dec 14 2008
%F A119881 a(n) = 2^n |E(n,-1)| where E(n,x) are the Euler polynomials. - _Peter Luschny_, Jan 25 2009
%F A119881 The odd part of a(n) = numerator(Euler(n,2)/2) = 1, 3, 1, 9, 1, 3, 1, 33, 1, -27, 1, 699, ... (compare A143074). - _Peter Luschny_, Nov 23 2012
%F A119881 G.f.: 1/Q(0), where Q(k) = 1 - 2*x - x*(k+1)/(1+x*(k+1)/Q(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Apr 19 2013
%F A119881 G.f.: 1/Q(0), where Q(k) = 1 - 4*x + x*(k+1)/(1-x*(k+1)/Q(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Apr 20 2013
%F A119881 a(n) = 2^(n+1)*(zeta[-n]*(2^(n+1)-1)+1). - _Peter Luschny_, Jul 16 2013
%F A119881 E.g.f.: 2/Q(0), where Q(k) = 1 + 2^k/( 1 - 2*x/( 2*x - 2^k*(k+1)/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Dec 16 2013
%F A119881 a(n) = 2^(n+1)*(1+(-1)^n*(2^(n+1)-1)*Bernoulli(n+1)/(n+1)). - _Vladimir Reshetnikov_, Oct 21 2015
%p A119881 a := proc(n) add(binomial(n,k)*bernoulli(k,1)*2^(n+k)/(n-k+1),k=0..n) end: # _Peter Luschny_, Dec 14 2008
%p A119881 a := n -> 2^n*abs(euler(n,-1)):  # _Peter Luschny_, Jan 25 2009
%p A119881 P := proc(n,x) option remember; if n = 0 then 1 else
%p A119881    (n*x+2*(1-x))*P(n-1,x)+x*(1-x)*diff(P(n-1,x),x);
%p A119881    expand(%) fi end:
%p A119881 A119881 := n -> subs(x=-1,P(n,x)):
%p A119881 seq(A119881(n), n=0..27);  # _Peter Luschny_, Mar 07 2014
%t A119881 Table[2^(n+1) (Zeta[-n] (2^(n+1)-1)+1), {n,0,27}] (* _Peter Luschny_, Jul 16 2013 *)
%t A119881 Range[0, 30]! CoefficientList[Series[Exp[3 x] Sech[x], {x, 0, 30}], x] (* _Vincenzo Librandi_, Mar 08 2014 *)
%o A119881 (Sage)
%o A119881 def skp(n, x):
%o A119881     A = lambda k: 0 if (k+1)%4 == 0 else (-1)^((k+1)//4)*2^(-(k//2))
%o A119881     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 A119881 A119881 = lambda n: skp(n,3)
%o A119881 [A119881(n) for n in (0..27)]  # _Peter Luschny_, Nov 23 2012
%o A119881 (PARI) my(x='x+O('x^66)); Vec(serlaplace(exp(3*x)/cosh(x))) \\ _Joerg Arndt_, Apr 20 2013
%o A119881 (Magma)
%o A119881 EulerPoly:= func< n,x | (&+[ (&+[ (-1)^j*Binomial(k,j)*(x+j)^n : j in [0..k]])/2^k: k in [0..n]]) >;
%o A119881 A119881:= func< n| (-2)^n*EulerPoly(n,-1) >;
%o A119881 [A119881(n): n in [0..40]]; // _G. C. Greubel_, Jun 07 2023
%Y A119881 Cf. A027641, A027642, A119879, A119880, A153641.
%K A119881 easy,sign
%O A119881 0,2
%A A119881 _Paul Barry_, May 26 2006