login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A009006 E.g.f. 1 + tan(x). 18
1, 1, 0, 2, 0, 16, 0, 272, 0, 7936, 0, 353792, 0, 22368256, 0, 1903757312, 0, 209865342976, 0, 29088885112832, 0, 4951498053124096, 0, 1015423886506852352, 0, 246921480190207983616, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

If b(0)=1 and b(n+1) = -sum(u(k)*binomial(n,k)*2^(n-k-1),k=0..n-1) then a(n) = abs(b(n)) (in fact b(n) = 1,1,0,-2,0,16,0,-272,...). - Robert FERREOL, Dec 30 2006

Sum_{k, 0<=k<=n}A075263(n,k)*2^k = 1,-1,0,2,0,-16,0,272,0,-7936,0,...for n=0, 1, 2, 3, 4, ...respectively . - Philippe Deléham, Aug 20 2007

REFERENCES

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1997; See Exercise 1.41(d).

LINKS

Table of n, a(n) for n=0..26.

Kwang-Wu Chen, An Interesting Lemma for Regular C-fractions, J. Integer Seqs., Vol. 6, 2003.

FORMULA

Let b(n) be a(n) shifted one place to the left with b(2+4k)=-a(3+4k), k=0, 1, .. Then b(n) is the expansion of sech(x)^2. - Mario Catalani (mario.catalani(AT)unito.it), Feb 08 2003

g(x)=x+x^2-2*x^4+16*x^6-272*x^8+... satisfies g(x/(1+2x))=-g(-x).

E.g.f.: 1+tan(x).

E.g.f. exp(x)*sech(x) is 1,1,0,-2,0,16,0,-272,... (A155585) - Paul Barry, Mar 15 2006

a(n)= 2^n*abs(Euler(n,0)) where Euler(n,x) is the n-th Eulerian polynomial. - Robert FERREOL, Dec 30 2006

E.g.f. -log(cos(x)), for n>0. - Vladimir Kruchinin, Aug 09 2010

a(n) = sum_{k=1..n}sum_{j=0..k}(-1)^(floor(n/2)+j+1)*binomial(n+1,k-j)*j^n for n>0. - Peter Luschny, Jul 23 2012

From Sergei N. Gladkovskii, Oct 25 2012 - Dec 20 2013: (Start) Continued fractions:

G.f.: 1 + x/T(0) where T(k) = 1 - (k+1)*(k+2)*x^2/T(k+1).

E.g.f.: 1 + tan(x) = 1+x/(U(0)-x) where U(k)= 4*k+1 + x/(1+x/(4*k+3 - x/(1- x/U(k+1)))).

E.g.f.: 1+tan(x) = 1 - 3*x/((U(0) + 3*x^2) where U(k) = 64*k^3 + 48*k^2 - 4*k*(2*x^2+1) - 2*x^2 - 3 - x^4*(4*k-1)*(4*k+7)/U(k+1).

E.g.f.: 1+x*G(0) where G(k) = 1 - x^2/(x^2 - (2*k+1)*(2*k+3)/G(k+1)).

G.f.: 1 + x/G(0) where G(k) = 1 - 2*x^2*(4*k^2+4*k+1)-4*x^4*(k+1)^2*(4*k^2+8*k+3) /G(k+1).

G.f.: 1 + x*Q(0) where Q(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - 1/Q(k+1)).

G.f.: Q(0) where Q(k) = 1 + x*(k+1)/(x*(k+1)+1/(1- x*(k+1)/(x*(k+1) - 1/Q(k+1)))).

E.g.f.: 2 - 1/Q(0) where Q(k) = 1 + x/(4*k+1 - x/(1 - x/(4*k+3 + x/Q(k+1)))). (End)

a(n) ~ 2*n!*(2/Pi)^(n+1) if n is odd. - Vaclav Kotesovec, Jun 01 2013

a(n) = i^(n+1) * 2^n * ((-1)^n -1) * (2^(n+1)-1) * Bernoulli(n+1)/(n+1), n>0. - Benedict W. J. Irwin, May 27 2016

MAPLE

u:=proc(n) if n=0 then 1 else -add(u(k)*binomial(n, k)/2*2^(n-k), k=0..n-1) fi end; seq(u(n), n=0..15); # Robert FERREOL, Dec 30 2006

MATHEMATICA

a[m_] := Abs[Sum[(-2)^(m-k) k! StirlingS2[m, k], {k, 0, m}]]; Table[a[i], {i, 0, 20}] (* Peter Luschny, Apr 29 2009 *)

PROG

(PARI) a(n)=if(n<1, n==0, n!*polcoeff(tan(x+x*O(x^n)), n))

(Sage)

def A009006(n) :

    if n == 0 : return 1

    return add(add((-1)^(n//2+j+1)*binomial(n+1, k-j)*j^n for j in (0..k)) for k in (1..n))

[A009006(n) for n in (0..26)] # Peter Luschny, Jul 23 2012

CROSSREFS

A000182(n)=a(2n-1).

Sequence in context: A111978 A146558 A025600 * A155585 A236755 A057375

Adjacent sequences:  A009003 A009004 A009005 * A009007 A009008 A009009

KEYWORD

nonn

AUTHOR

R. H. Hardin

EXTENSIONS

Reformatted Mar 15 1997

Definition corrected by Joerg Arndt, Apr 29 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 23 13:19 EST 2018. Contains 299581 sequences. (Running on oeis4.)