[seqfan] Surprising Patterns in Tangent and Secant Numbers
Paul D Hanna
pauldhanna at juno.com
Tue Nov 10 05:57:30 CET 2009
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
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