a possibly interesting connection involving OEIS A343846.

[Davide Rotondo]

A343846 is defined by

a(n) = binomial(2n,n) * 2^n * |Euler(n,1/2) - Euler(n,0)|

and begins

2, 6, 40, 350, 4032, 56364, 933504, 17824950, 385848320, ...

I found that these numbers can be generated very naturally using the Euler/Seidel-Entringer zig-zag (boustrophedon) construction.

For each n, start with a row consisting of n+1 ones:

n = 1:
1 1

n = 2:
1 1 1

n = 3:
1 1 1 1

and so on.

Then repeatedly apply the usual alternating cumulative-sum operation of the boustrophedon construction, changing the direction at each step. After n steps, take the appropriate outer edge, alternating between the left and right edge.

This gives:

2
6
40
350
4032
56364
933504
17824950
385848320
9334057876
...

which agrees with A343846.

For example:

1 1
0 1 2
       ^
       2

and

1 1 1
0 1 2 3
6 6 5 3 0
^
6

and

1 1 1 1
0 1 2 3 4
10 10 9 7 4 0
0 10 20 29 36 40 40
                                ^
                               40

and

1 1 1 1 1
0 1 2 3 4 5
15 15 14 12 9 5 0
0 15 30 44 56 65 70 70
350 350 335 305 261 205 140 70 0
^
350

This appears to have a direct explanation through the Euler triangle. The relevant identity is

T(n,k) = binomial(n,k) * A000111(n-k),

where A000111 is the Euler zigzag sequence.

Taking the central entry gives

T(2n,n) = binomial(2n,n) * A000111(n).

Using the standard relation

A000111(n) = 2^n |Euler(n,1/2) - Euler(n,0)|,

we obtain exactly

T(2n,n) = A343846(n).

Thus the construction can apparently be summarized as

finite boustrophedon construction from (1,1,...,1)
→ central Euler-triangle entry
→ binomial(2n,n) A000111(n)
→ A343846.

I realize that the underlying objects are classical, and the connection between Euler zigzag numbers, the Seidel-Entringer triangle, and the Euler triangle is well known. I am therefore NOT claiming that the sequence or these identities are new.

What I have not been able to find is an explicit description of A343846 using this particular finite construction: for each n, start with n+1 ones, perform n alternating cumulative-sum steps, and select the appropriate outer edge.

I would be very interested to know whether this exact construction has already appeared in the literature or in OEIS, or whether it gives a useful new combinatorial/algorithmic interpretation of A343846.

In particular, I would also like to know whether this construction can be generalized by replacing the initial row

1,1,...,1

with other simple finite sequences, potentially producing other interesting OEIS sequences or known diagonals of Euler/Seidel-Entringer-type triangles.

Any references, corrections, or comments on the exact status of this observation would be very welcome.

Davide