An identity related to the second-order Eulerian numbers.

Question

Recently, some of the remarkable properties of second-order Eulerian numbers $ \left\langle\!\!\left\langle n\atop k\right\rangle\!\!\right\rangle$ A340556 have been proved on MSE [ a , b , c ].

But there are also other notable identities to which these numbers lead. For instance, we also noticed the following identity, which we haven't seen elsewhere (reference?):

$$ \sum_{j=0}^{k} \binom{n-j}{n-k} \left\langle\!\! \left\langle n\atop j\right\rangle\!\! \right\rangle \,=\, \sum_{j=0}^k (-1)^{j+k} \binom{n+k}{n+j} \left\{ n+j \atop j\right \} \quad ( n \ge 0) $$

If we use the notation $ \operatorname{W}_{n, k} $ for these numbers, we can also introduce the corresponding polynomials.

$$ \operatorname{W}_{n}(x) = \sum_{k=0}^n \operatorname{W}_{n, k} x^k \quad ( n \ge 0) $$

So far so routine, but then came the surprise:

$$ 3^n W_n\left(-\frac13\right) \, = \, 2^n \left\langle\!\!\left\langle - \frac{1}{2} \right\rangle\!\!\right\rangle_n \quad ( n \ge 0) $$

On the right side are the numbers we recently asked about their combinatorial significance!

Should I trust this strange equation?

Answer 1

In trying to verify the identity

$$\sum_{j=0}^{k} {n-j \choose n-k} \left\langle\!\! \left\langle n\atop j \right\rangle\!\! \right\rangle = \sum_{j=0}^k (-1)^{j+k} {n+k \choose n+j} \left\{ n+j \atop j\right \}$$

we quote from MSE the following identity

$$\left\langle\!\! \left\langle n\atop k \right\rangle\!\! \right\rangle = \sum_{j=0}^k (-1)^{k-j} {2n+1\choose k-j} {n+j\brace j}.$$

We get for the LHS

$$\sum_{j=0}^{k} {n-j \choose n-k} \sum_{p=0}^j (-1)^{j-p} {2n+1\choose j-p} {n+p\brace p} \\ = \sum_{p=0}^k {n+p\brace p} \sum_{j=p}^k (-1)^{j-p} {2n+1\choose j-p} {n-j\choose n-k}.$$

The inner sum is $$\sum_{j=0}^{k-p} (-1)^j {2n+1\choose j} {n-j-p\choose n-k} \\ = \sum_{j=0}^{k-p} (-1)^j {2n+1\choose j} {n-j-p\choose k-p-j} \\ = [z^{k-p}] (1+z)^{n-p} \sum_{j=0}^{k-p} (-1)^j {2n+1\choose j} \frac{z^j}{(1+z)^j}.$$

Here the coefficient extractor enforces the upper limit of the sum and we may extend $j$ to infinity:

$$[z^{k-p}] (1+z)^{n-p} \sum_{j\ge 0} (-1)^j {2n+1\choose j} \frac{z^j}{(1+z)^j} \\ = [z^{k-p}] (1+z)^{n-p} \left(1-\frac{z}{1+z}\right)^{2n+1} = [z^{k-p}] \frac{1}{(1+z)^{n+p+1}} \\ = (-1)^{k-p} {k-p+n+p\choose n+p} = (-1)^{k-p} {n+k\choose n+p}.$$

Introducing the leading term,

$$\sum_{p=0}^k (-1)^{k+p} {n+k\choose n+p} {n+p\brace p}$$

This is the claim.

As for the polynomials we find

$$\sum_{k=0}^n x^k \sum_{j=0}^{k} {n-j \choose n-k} \left\langle\!\! \left\langle n\atop j \right\rangle\!\! \right\rangle = \sum_{j=0}^n \left\langle\!\! \left\langle n\atop j \right\rangle\!\! \right\rangle \sum_{k=j}^n {n-j\choose n-k} x^k \\ = \sum_{j=0}^n \left\langle\!\! \left\langle n\atop j \right\rangle\!\! \right\rangle x^j \sum_{k=0}^{n-j} {n-j\choose n-j-k} x^k = \sum_{j=0}^n \left\langle\!\! \left\langle n\atop j \right\rangle\!\! \right\rangle x^j \sum_{k=0}^{n-j} {n-j\choose k} x^k \\ = \sum_{j=0}^n \left\langle\!\! \left\langle n\atop j \right\rangle\!\! \right\rangle x^j (1+x)^{n-j} = (1+x)^n \sum_{j=0}^n \left\langle\!\! \left\langle n\atop j \right\rangle\!\! \right\rangle \frac{x^j}{(1+x)^j}.$$

Multiplying by $3^n$ and evaluating at $x=-1/3$ we obtain

$$3^n \frac{2^n}{3^n} \sum_{j=0}^n \left\langle\!\! \left\langle n\atop j \right\rangle\!\! \right\rangle \frac{(-1/3)^j}{(2/3)^j} = 2^n \sum_{j=0}^n \left\langle\!\! \left\langle n\atop j \right\rangle\!\! \right\rangle \left(-\frac{1}{2}\right)^j$$

as claimed.

Remark. Maybe we can pause here for a few days. Thanks!

Answer 2

A sketch in three steps: First, we need a definition for $ \operatorname{W}_{n}(x)$. With this, we do not want to assume the second-order Eulerian numbers. So we take the RHS of the identity.

The next and primary step is to show that in general

$$ \operatorname{W}_{n}(x) = (1 + x)^n \left\langle \! \left\langle \frac{x}{1+x} \right\rangle \! \right\rangle _n. $$

Finally, plug in $ x = -\frac13 $, et voilà, $$ \operatorname{W}_{n}\left( - \frac13 \right) = \left( \frac23 \right)^n \left\langle \! \left\langle -\frac12 \right\rangle \! \right\rangle _n .$$

The choice of the notation $ \operatorname{W}_{n}(x) $ was not accidental. We are speaking here about the Ward polynomials. Their coefficients are the Ward numbers which be found at A269939.