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A000810 Expansion of e.g.f. (sin x + cos x)/cos 3x. 3
1, 1, 8, 26, 352, 1936, 38528, 297296, 7869952, 78098176, 2583554048, 31336418816, 1243925143552, 17831101321216, 825787662368768, 13658417358350336, 722906928498737152, 13551022195053101056 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

R. J. Mathar, Table of n, a(n) for n = 1..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.dps = 32; mp.pretty = True

def A000810(n): return abs(3^n*2^(n+1)*lerchphi(-1, -n, 1/3))

[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.

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

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Last modified April 25 19:05 EDT 2017. Contains 285426 sequences.