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A009006 Expansion of e.g.f.: 1 + tan(x). 26
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, 70251601603943959887872, 0, 23119184187809597841473536, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

If b(0)=1 and b(n+1) = -Sum_{k=0..n-1} u(k)*binomial(n,k)*2^(n-k-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..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

G. C. Greubel, Table of n, a(n) for n = 0..451

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

a(0) = a(1) = 1; a(n) = -2 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1). - Ilya Gutkovskiy, Jul 05 2020

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 *)

A009006[n_] := Cos[Pi (n-1) / 2] (4^(n+1) - 2^(n+1)) * BernoulliB[n+1] / (n+1); a[0] := 1; Table[A009006[n], {n, 0, 30}] (* Peter Luschny, Jun 14 2021 *)

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

(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1 + Tan(x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jul 21 2018

CROSSREFS

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

Sequence in context: A111978 A146558 A025600 * A155585 A350972 A236755

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

Terms a(26) onward added by G. C. Greubel, Jul 21 2018

STATUS

approved

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Last modified February 5 19:42 EST 2023. Contains 360087 sequences. (Running on oeis4.)