OFFSET
1,3
COMMENTS
Terms are of form 10k+1.
LINKS
Eric Weisstein's World of Mathematics, Euler Polynomial.
FORMULA
From Peter Bala, Jul 26 2013: (Start)
Conjectural e.g.f. with offset 0 (checked up to a(14)): 1/3*(2 - cos(x)^2 + 2*cos(x)^4)/cos(x)^3 = 1 + x^2/2! + 21*x^4/4! + 461*x^6/6! + .... (End)
G.f.: 1/(Q(0)*3*x) + 2/(3*x^2*(1+x)) - 2/(3*x^2) + 1/(3*x), where Q(k) = 1 - x*(k+1)^2/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Sep 19 2013
a(n) = (2n)! * [x^(2n)] 1/3-2*sin(x)/(3*tan(2*x)). - Vladimir Kruchinin, Apr 08 2015
Conjecture: a(n) = -1/3*(-4)^n*E(2*n,-1/2), where E(n,x) is the n-th Euler polynomial. - Peter Bala, Sep 25 2016
MATHEMATICA
max = 28; CoefficientList[Series[1/3-2*Sin[x]/(3*Tan[2*x]), {x, 0, max}], x^2] * Range[0, max, 2]! // Rest (* Jean-François Alcover, Apr 08 2015, after Vladimir Kruchinin *)
PROG
(PARI) a(n)=(-1)^(n+1)/3/(2*n+1)*sum(k=0, n, 16^k*bernfrac(2*k)*binomial(2*n+1, 2*k))
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Benoit Cloitre, Oct 19 2002
STATUS
approved