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 A009006 Expansion of e.g.f.: 1 + tan(x). 19
 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 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 (MAGMA) m:=30; R:=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 A236755 A057375 Adjacent sequences:  A009003 A009004 A009005 * A009007 A009008 A009009 KEYWORD nonn AUTHOR 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 October 23 14:54 EDT 2019. Contains 328345 sequences. (Running on oeis4.)