[seqfan] Re: Surprising Patterns in Tangent and Secant Numbers

[Paul D Hanna]

Franklin (and SeqFans), 
      I appreciate your consideration, but perhaps you dismiss the situation too quickly. 
 
The surprise is not that the e.g.f. exp(x)/cosh(x) = 1 + tanh(x) 
has zeros every other term, but that the alternating zeros 
also occur under the transformation of the o.g.f.: 
   F(x) = 1 + x - 2*x^3 + 16*x^5 - 272*x^7 + 7936*x^9 +...
described by:    
 
> Now if we define B(x) by: 
>   B(x) = F(x/B(x))  and F(x) = B(x*F(x)) 
> so that
>   B(x) = x/Series_Reversion(x*F(x)) 
>        = 1 + x - x^2 + 3*x^4 - 38*x^6 + 947*x^8 - 37394*x^10 +...
>   
> then we observe that B(x) is a function that also has 
> alternating zero coefficients, but at odd positions (>1):
> A157308 = [1,1,-1,0,3,0,-38,0,947,0,-37394,0,2120190,0,...].
  
The re-emergence of alternating zeros occurring at opposite positions is not trivial; 
it is unique under that transformation (given that the function begins 1+x+...). 
  
Example: 
x/Series_Reversion[x/(1+x^2) +x^2] = 1 + x - 2*x^2 + 5*x^3 - 16*x^4 + 59*x^5 - 233*x^6 +... 
In this example, the resultant series is neither odd nor even under the transformation, 
even though we started with x/(1+x^2) +x^2 = x + x^2 - x^3 + x^5 - x^7 + x^9 +...  
which clearly has alternating zero coefficients after the initial terms. 
  
 
I believe that the behavior is directly linked to the following identity: 
 
x/serreverse(x*serlaplace(exp(x)/cosh(x))) 
  
= x + x/serreverse(x*serlaplace(1/cosh(x)))
  
= 1 + x - x^2 + 3*x^4 - 38*x^6 + 947*x^8 - 37394*x^10 +...
  
Prove this identity and the other observations (I think) should follow by logical deduction.  
        Paul 
 
---------- Original Message ----------
From: franktaw at netscape.net
To: seqfan at list.seqfan.eu
Subject: [seqfan] Re: Surprising Patterns in Tangent and Secant Numbers
Date: Tue, 10 Nov 2009 04:51:14 -0500

You're making this harder than it really is.  A function has zeros in 
odd positions iff it is an even function (that is, f(-x) = -f(x)), and 
zeros in even positions if it is an odd function (f(-x) = f(x)).  So 
the tangent numbers have zeros in even positions except the constant 
term if

exp(x)/cosh(x) - 1 = (exp(x) - exp(-x))/(exp(x) + exp(-x))

is odd; and by inspection, it is.

I'm less clear about what is happening in A157310, but I would 
definitely approach it by trying to show that C(x) - 1 is an odd 
function.

Franklin T. Adams-Watters