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A119881 Expansion of e.g.f. exp(3*x)*sech(x). 5
1, 3, 8, 18, 32, 48, 128, 528, 512, -6912, 2048, 357888, 8192, -22351872, 32768, 1903822848, 131072, -209865080832, 524288, 29088886161408, 2097152, -4951498048929792, 8388608, 1015423886523629568, 33554432, -246921480190140874752, 134217728, 70251601603944228323328 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Transform of 3^n under the matrix A119879.

Also the Swiss-Knife polynomials A153641 evaluated at x=3. - Peter Luschny, Nov 23 2012

LINKS

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

FORMULA

a(n) = sum{k=0..n, A119879(n,k)3^k}.

a(n) = Sum(binomial(n,k)*B(k,1)*2^(n+k)/(n-k+1), k=0..n). 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

a(n) = 2^n |E(n,-1)| where E(n,x) are the Euler polynomials. - Peter Luschny, Jan 25 2009

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

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

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

a(n) = 2^(n+1)*(zeta[-n]*(2^(n+1)-1)+1). - Peter Luschny, Jul 16 2013

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

a(n) = 2^(n+1)*(1+(-1)^n*(2^(n+1)-1)*Bernoulli(n+1)/(n+1)). - Vladimir Reshetnikov, Oct 21 2015

MAPLE

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

a := n -> 2^n*abs(euler(n, -1)):  # Peter Luschny, Jan 25 2009

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

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

   expand(%) fi end:

A119881 := n -> subs(x=-1, P(n, x)):

seq(A119881(n), n=0..27);  # Peter Luschny, Mar 07 2014

MATHEMATICA

Table[2^(n+1) (Zeta[-n] (2^(n+1)-1)+1), {n, 0, 27}] (* Peter Luschny, Jul 16 2013 *)

Range[0, 30]! CoefficientList[Series[Exp[3 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))

A119881 = lambda n: skp(n, 3)

[A119881(n) for n in (0..27)]  # Peter Luschny, Nov 23 2012

(PARI) x='x+O('x^66); Vec(serlaplace(exp(3*x)/cosh(x))) \\ Joerg Arndt, Apr 20 2013

CROSSREFS

Cf. A119880.

Sequence in context: A288249 A004210 A247022 * A184636 A075342 A083726

Adjacent sequences:  A119878 A119879 A119880 * A119882 A119883 A119884

KEYWORD

easy,sign

AUTHOR

Paul Barry, May 26 2006

STATUS

approved

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Last modified June 27 15:12 EDT 2017. Contains 288790 sequences.