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Assume $a\in\mathbb{N}$. Among the families of sequences studied by Nicolaas de Bruijn (Asymptotic Methods in Analysis, 1958), let's focus on the (modified) $$\hat{S}(2a,n)=\frac1{n+1}\sum_{k=0}^{2n}(-1)^{n+k}\binom{2n}k^{2a}.$$ An all-familiar fact states: the Catalan number $C_n=\frac1{n+1}\binom{2n}n$ is odd iff $n=2^m-1$.

Encouraged by a positive response of Fedor Petrov to my earlier MO question regarding the case $a=2$, I decided to beef-up the quest into a generalization.

QUESTION. Is this true? $\hat{S}(2a,n)$ is odd iff $n=2^m-1$ for some $m\in\mathbb{Z_{\geq0}}$.

Remark. Notice that in the case $a=1$, we have $\hat{S}(2,n)=C_n$.

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  • $\begingroup$ @FedorPetrov: the sum run through $0$ to $2n$. Why do you expect it to drop to $n$? $\endgroup$ Feb 7 at 14:22
  • $\begingroup$ Because I am confused: the sum in RHS in the main identity for $a=2$ was for $k$ from 0 to $n$. $\endgroup$ Feb 7 at 14:43
  • $\begingroup$ @FedorPetrov: no, it was up to $2n$. $\endgroup$ Feb 7 at 15:22
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    $\begingroup$ Calkin showed (see theorem 1 here matwbn.icm.edu.pl/ksiazki/aa/aa86/aa8612.pdf) that $C_n|\hat{S}(2a,n)$ so it only remains to check indeed that for $n=2^m-1$ the sum is odd. $\endgroup$
    – Vlad Matei
    Feb 7 at 19:25
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I think, yes and it may be proved by the same way as for $a=2$. We note that $$ C:=\sum_{k=0}^{2n}(-1)^k\binom{2n}k^{2a}=\left[x_1^{2n}\ldots x_{2a}^{2n}\right] (x_1-x_2)^{2n}(x_2-x_3)^{2n}\ldots (x_{2a-1}-x_{2a})^{2a}(x_{2a}+x_1)^{2n}\\= \left[x_1^{2n}\ldots x_{2a}^{2n}\right]F(x_1,\ldots,x_{2a}), $$ where $$F(x_1,\ldots,x_{2a})=\prod_{j=-(n-1)}^n(x_1-x_2-j)(x_3-x_2-j)(x_3-x_4-j)\ldots(x_{2a-1}-x_{2a}-j)(x_{2a}+x_1-2n-j).$$ We denote $A=\{0,1,\ldots,2n\}$ and express the above coefficient via the values of $F$ on $A\times A\times \ldots \times A$ applying the Combinatorial Nullstellensatz formula. It is not hard to see (pretty analogous to $a=2$ case which was considered in details in my answer to the previous question) that if $F(x_1,\ldots,x_{2a})\ne 0$ for $x_i\in A$ then we must have $x_1=x_{2a-1}=0$, $x_{2a}=n$. Then $F(x_1,\ldots,x_{2n})$ is divisible by $((2n)!)^{2a}$ and the denominator in the CN formula equals $(-1)^{\sum x_i}\prod_{i=1}^{2a}x_i!(2n-x_i)!=((2n)!)^2 (n!)^2(-1)^{n+\sum_{i=2}^{2a-1}x_i}\prod_{i=2}^{2a-1}x_i!(2n-x_i)!$. It is immediately seen that $C$ is divisible by ${2n\choose n}$ (that already implies that $\hat{S}(2a,n)$ is divisible by $C_n$, as was earlier proved by Calkin by a different argument, see Vlad's reference). When $n=2^m-1$, using the fact that the product of $2n$ consecutive integers like $\prod_{j=-(n-1)}^n(x_1-x_2-j)$ is divisible by $2\cdot (2n)!$ unless $x_1-x_2\in \{n,n+1\}$ (again, see the explanation in the previous answer) and that $(2n)!/(x_i!(2n-x_i)!)$ is even when $x_i$ is odd, we see that the unique odd summand in our sum corresponds to $x_1=x_3=\ldots=x_{2a-3}=0$, $x_2=x_4=\ldots=x_{2a-2}=n+1$.

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