[seqfan] Surprising Patterns in Tangent and Secant Numbers

[Paul D Hanna]

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. 
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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,...]. 
 
Instead of the e.g.f., we will look at the o.g.f.:
F(x) = 1 + x - 2*x^3 + 16*x^5 - 272*x^7 + 7936*x^9 +...
  
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,...].
 
I find it surprising that both sequences A155585 and A157308 
have alternating zero terms under the given transformation. 
A proof would be nice. 
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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. 
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Together, these patterns are doubly surprising to me, and 
I would like a proof that the patterns continue to hold. 
What I have so far is convincing emperical evidence. 
 
  
These patterns also make one suspect that some nice formula exists 
for the terms in A157308 and A157310, the unsigned bisections being:
A158119 = [1,1,3,38,947,37394,2120190,162980012,16330173251,...].
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A FORMULA FOR A158119.  
Consider sequence A000364, which is the Euler (or Secant) numbers, defined by 
E.g.f.: 1/cos(x).
   
Let E(x) = o.g.f. of A000364 when ignoring the zero terms:
E(x) = 1 + x + 5*x^2 + 61*x^3 + 1385*x^4 + 50521*x^5 +...
 
Now let D(x) be the o.g.f. of A158119: 
D(x) = 1 + x + 3*x^2 + 38*x^3 + 947*x^4 + 37394*x^5 +...
 
Then emperical evidence suggests that:
  D(x) = E(x/D(x)^2)  and E(x) = D(x*E(x)^2) 
so that: 
  D(x)^2 = x/Series_Reversion(x*E(x)^2)
  E(x)^2 = x*Series_Reversion(x/D(x)^2)
 
This implies that the sequences are related by:
A158119(n) = [x^n] E(x)^(1-2n)/(1-2n)
A000364(n) = [x^n] D(x)^(2n+1)/(2n+1) 
 
Again, a proof of the above would be nice. 
And perhaps there is some simpler formula for A158119?  
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What makes the above formulas difficult is that I do not know 
an elementary expression for the O.g.f.s of A155585 and A000364, 
only the E.g.f.s exp(x)/cosh(x) and 1/cos(x), respectively. 
 
If someone knows nice expressions for the o.g.f.s of either
 A155585 or A000364, it would be helpful. 
      
Thanks for any comments or brilliant insights,
    Paul