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A008990 Numerators of sequence having sqrt(cos(x)) as e.g.f. (even-indexed coefficients only). 1
1, -1, -1, -19, -559, -29161, -2368081, -276580459, -43947282079, -9118829535121, -2394495729300961, -776228170682260099, -304471093666800990799, -142128398853646068197881, -77865168574139358455774641, -49474260304294496117945326939 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
REFERENCES
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 191.
H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 206.
LINKS
FORMULA
E.g.f.: sqrt(cos(sqrt(2)*x)) = 1 - x^2/2! - x^4/4! - 19x^6/6! -... - Ralf Stephan, Mar 03 2005
a(n) = sum(sum(binomial(k,j)*2^(n+2-2*k-j)*sum(binomial(j,i)*(j-2*i)^(2*n), i=0..floor((j-1)/2))*(-1)^(n+j+1), j=1..k)*C(k-1), k=1..2*n), n>0, C(n) - Catalan numbers (A000108). - Vladimir Kruchinin, Sep 10 2010
G.f.: 2/G(0) where G(k) = 2 - 4*x*(k+1)*(2*k-1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 12 2013
G.f.: Q(0), where Q(k) = 1 - x*(k+1)*(2*k-1)/( x*(k+1)*(2*k-1) + 2/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2013
a(n) = (1/(4*n))*Sum_{k=1..n} C(2*n,2*k)*(-1)^(k)*2^(3*k)*(2^(2*k)-1)*B(2*k)*a(n-k), a(0)=1, where B(n) is Bernoulli numbers. - Vladimir Kruchinin, Jun 23 2015.
The odd terms of EllipticE(x,2) act as a g.f. for 2^n*a(n)/(2*n+1)!. - Benedict W. J. Irwin, Jun 06 2016
a(n) ~ -2^(5*n - 1) * n^(2*n - 1) / (Pi^(2*n - 1/2) * exp(2*n)). - Vaclav Kotesovec, Jun 11 2016
O.g.f. as a continued fraction of Stieltjes type: 1/(1 + x/(1 - 2*x/(1 - 9*x/(1 - 20*x/(1 - ... - (2*n^2 + n - 1)*x/(1 - ... )))))) = 1 - x - x^2 - 19*x^3 - 559*x^4 - .... Follows from Wall, equation 53.11, p. 206 with k = - 1/2. - Peter Bala, Apr 11 2022
MATHEMATICA
n = 32; Partition[ CoefficientList[ Series[ Sqrt[Cos[Sqrt[2]*x]], {x, 0, n}], x]*Range[0, n]!, 2][[All, 1]] (* Jean-François Alcover, Aug 30 2011 *)
Table[SeriesCoefficient[Series[EllipticE[x, 2], {x, 0, 41}], 2 n + 1] (2 n + 1)!/2^n, {n, 0, 20}] (* Benedict W. J. Irwin, Jun 06 2016 *)
PROG
(Maxima) C(n):=1/(n+1)*binomial(2*n, n); a(n):=sum(sum(binomial(k, j) *2^(n+2-2*k-j)*sum(binomial(j, i)*(j-2*i)^(2*n), i, 0, floor((j-1)/2))*(-1)^(n+j+1), j, 1, k)*C(k-1), k, 1, 2*n); /* Vladimir Kruchinin, Sep 10 2010 */
(Maxima) a(n):=if n=0 then 1 else 1/(4*n)*sum(binomial(2*n, 2*k)*(-1)^(k)*2^(3*k)*(2^(2*k)-1)*bern(2*k)*a(n-k), k, 1, n); /* Vladimir Kruchinin, Jun 23 2015 */
CROSSREFS
Cf. A027641.
Denominators are in A000079.
Sequence in context: A278184 A035278 A092611 * A012845 A284111 A142023
KEYWORD
sign,frac,easy,nice
AUTHOR
STATUS
approved

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Last modified July 18 00:08 EDT 2024. Contains 374377 sequences. (Running on oeis4.)