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A119880 E.g.f. exp(2x)*sech(x). 8
1, 2, 3, 2, -3, 2, 63, 2, -1383, 2, 50523, 2, -2702763, 2, 199360983, 2, -19391512143, 2, 2404879675443, 2, -370371188237523, 2, 69348874393137903, 2, -15514534163557086903, 2, 4087072509293123892363, 2, -1252259641403629865468283, 2, 441543893249023104553682823 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Transform of 2^n under the matrix A119879.

Also the Swiss-Knife polynomials A153641 evaluated at x=2. - Peter Luschny, Nov 23 2012

LINKS

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

Paul Barry, Eulerian-Dowling Polynomials as Moments, Using Riordan Arrays, arXiv:1702.04007 [math.CO], 2017.

FORMULA

a(n) = Sum_{k=0..n} A119879(n,k) * 2^k.

G.f.: 1/U(0) where U(k)=  1 - x - x*(k+1)/(1 + x*(k+1)/U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Oct 14 2012

G.f.: 1/Q(0), where Q(k)= 1 - 3*x + x*(k+1)/(1-x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 20 2013

G.f.: x/(1-x)/Q(0) + 1/(1-x), where Q(k)= 1 - x + x^2*(k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 27 2013

a(n) = 2*(1+zeta(-n)*(2^n-1)+2^(2*n+1)*zeta(-n,3/4)). - Peter Luschny, Jul 16 2013

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) ); (continued fraction). - Sergei N. Gladkovskii, Nov 11 2013

E.g.f.: 2/Q(0), where Q(k) = 1 + 3^k/( 1 - x/( x - 3^k*(k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 16 2013

MAPLE

A119880_list := proc(n) local S, A, m, k;

A := array(0..n-1, 0..n-1); S := NULL;

for m from 0 to n-1 do

   A[m, 0] := (-2)^m*euler(m, 0);

   for k from m-1 by -1 to 0 do

       A[k, m-k] := A[k+1, m-k-1] + A[k, m-k-1] od;

    S := S, A[0, m] od;

S end:

A119880_list(31); # Peter Luschny, Jun 15 2012

P := proc(n, x) option remember; if n = 0 then 1 else

  (n*x-(1/2)*(1-x))*P(n-1, x)+x*(1-x)*diff(P(n-1, x), x);

  expand(%) fi end:

A119880 := n -> (-1)^n*subs(x=-1, P(n, x)):

seq(A119880(n), n=0..30);  # Peter Luschny, Mar 07 2014

MATHEMATICA

Table[2 (1 + Zeta[-n] (2^n - 1) + 2^(2 n + 1) Zeta[-n, 3/4]), {n, 0, 30}] (* Peter Luschny, Jul 16 2013 *)

Range[0, 30]! CoefficientList[Series[Exp[2 x] Sech[x], {x, 0, 30}], x] (* Vincenzo Librandi, Mar 08 2014 *)

PROG

(Sage)

def skp(n, x):

    A = lambda k: 0 if (k+1)%4 == 0 else (-1)^((k+1)//4)*2^(-(k//2))

    return add(A(k)*add((-1)^v*binomial(k, v)*(v+x+1)^n for v in (0..k)) for k in (0..n))

A119880 = lambda n: skp(n, 2)

[A119880(n) for n in (0..30)]  # Peter Luschny, Nov 23 2012

CROSSREFS

Sequence in context: A108656 A164962 A251089 * A075019 A138960 A245553

Adjacent sequences:  A119877 A119878 A119879 * A119881 A119882 A119883

KEYWORD

easy,sign

AUTHOR

Paul Barry, May 26 2006

STATUS

approved

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Last modified August 17 22:32 EDT 2018. Contains 313817 sequences. (Running on oeis4.)