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

[franktaw at netscape.net]

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

-----Original Message-----
From: Paul D Hanna <pauldhanna at juno.com>

SeqFans,
    Consider the tangent numbers, which may be described by
E.G.F.: exp(x)/cosh(x) = 1 + x - 2*x^3/3! + 16*x^5/5! - 272*x^7/7! +...
and the the Euler (or Secant) numbers, defined by
E.g.f.: 1/cos(x) = 1 + x^2/2! + 5*x^4/4! + 61*x^6/6! + 1385*x^8/8! +

There are some surprising patterns hidden inside these sequences
that I would like to divulge here, for which I desire proofs
or some heuristic arguement that the patterns should hold.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

PATTERN I.
The tangent numbers in A155585 has alternating zero terms at even 
positions
(>0):
A155585 = [1,1,0,-2,0,16,0,-272,0,7936,0,-353792,0,...].

...
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

PATTERN II.
If we reverse the signs after the initial terms in A157308 like so:
A157310 = [1,1,1,0,-3,0,38,0,-947,0,37394,0,-2120190,0,...]
we get the o.g.f.:
G(x) = 1 + x + x^2 - 3*x^4 + 38*x^6 - 947*x^8 + 37394*x^10 -...

and, likewise, if we define C(x) by:
  C(x) = G(x/C(x))  and G(x) = C(x*G(x))
so that
  C(x) = x/Series_Reversion(x*G(x))
       = 1 + x - x^3 + 9*x^5 - 176*x^7 + 5693*x^9 - 272185*x^11 +...

then we observe that C(x) is a function that also has
alternating zero coefficients, back again at even positions (>0):
A157309 = [1,1,0,-1,0,9,0,-176,0,5693,0,-272185,0,18043492,0,...].

Again, I find it unexpected that both sequences A157310 and A157309
have alternating zero terms under the given transformation.
A proof would be nice.
...