A000810 Expansion of e.g.f. (sin x + cos x)/cos 3x.

1, 1, 8, 26, 352, 1936, 38528, 297296, 7869952, 78098176, 2583554048, 31336418816, 1243925143552, 17831101321216, 825787662368768, 13658417358350336, 722906928498737152, 13551022195053101056

Offset

0,3

Links

R. J. Mathar, Table of n, a(n) for n = 0..200

Formula

a(2n) = A000436(n).

(-1)^n*a(2n+1)=1-sum_{i=0,1,...,n-1} (-1)^i*binomial(2n+1,2i+1)*3^(2n-2i)*a(2i+1). - R. J. Mathar, Nov 19 2006

a(n) = | 3^n*2^(n+1)*lerchphi(-1,-n,1/3) |. - Peter Luschny, Apr 27 2013

a(n) ~ n!*2^(n+1)*3^(n+1/2)/Pi^(n+1) if n is even and a(n) ~ n!*2^(n+1)*3^n/Pi^(n+1) if n is odd. - Vaclav Kotesovec, Jun 25 2013

a(n) = (-1)^floor(n/2)*3^n*skp(n, 1/3), where skp(n,x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Apr 19 2014

Mathematica

CoefficientList[Series[(Sin[x]+Cos[x])/Cos[3*x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 25 2013 *)
Table[Abs[EulerE[n, 1/3]] 6^n, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 21 2015 *)

Prog

(Sage)
from mpmath import mp, lerchphi
mp.dps = 32; mp.pretty = True
def A000810(n): return abs(3^n*2^(n+1)*lerchphi(-1, -n, 1/3))
[int(A000810(n)) for n in (0..17)]  # Peter Luschny, Apr 27 2013
(PARI) x='x+O('x^66); v=Vec(serlaplace( (sin(x)+cos(x)) / cos(3*x) ) ) \\ Joerg Arndt, Apr 27 2013

Crossrefs

Cf. A007286, A007289.

(-1)^(n*(n-1)/2)*a(n) gives the alternating row sums of A225118. - Wolfdieter Lang, Jul 12 2017

Sequence in context: A089064 A240291 A203635 * A171740 A129663 A112646

Adjacent sequences: A000807 A000808 A000809 * A000811 A000812 A000813

Keyword

nonn

Author

N. J. A. Sloane

Status

approved