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Let $$a(n,m,k)=\frac{1}{n}\sum\limits_{j=0}^{n}[n+kj\geqslant 0]\binom{n}{j}\binom{n+kj}{j-1}(m-1)^{j-1}$$ Here square brackets denote Iverson brackets.

There are many sequences in the OEIS that are special cases of a given sequence family:

Let $$b(n,m,k,\ell)=\frac{1}{n}\sum\limits_{j=0}^{n}[n+kj-\ell\geqslant 0]\binom{n}{j}\binom{n+kj-\ell}{j-1}(m-1)^j$$

Let $c(n,m,k)$ be a sequence of the nonnegative integers such that $$c(m^p(mn+1),m,k)=c(m^p(mn+2),m,k)=\cdots=c(m^p(mn+m-1),m,k)=\sum\limits_{j=0}^{p+k}c(m^j n,m,k), c(0, m, k)=1$$

Let $s(n,m,k)$ be a sequence of the positive integers such that $$s(n,m,k)=\sum\limits_{j=0}^{m^n-1}c(j,m,k)$$

I conjecture that for $n\geqslant 0, m\geqslant 2, k\geqslant -1$ we have $$s(n,m,k)=a(n+1,m,k)$$ I also conjecture that for $n\geqslant 0, m\geqslant 2, k\in\mathbb{Z}$ we have $$s(n,m,k)=a(n+1,m,k)+[k<-1]([n<(|k|-1)]+\sum\limits_{i=2}^{|k|}(i-1)b(n+1,m,k,i))$$

Here is the PARI prog to verify these conjectures:

a(n, m, k) = (1/n)*sum(j=1, n, if(n + k*j >= 0, binomial(n, j)*binomial(n + k*j, j-1)*(m-1)^(j-1)))
b(n, m, k, l) = (1/n)*sum(j=1, n, if(n + k*j - l >= 0, binomial(n, j)*binomial(n + k*j - l, j-1)*(m-1)^j))
c(n, m, k) = if(n==0, 1, my(A=valuation(n, m), B=n\m^(A+1)); sum(j=0, A+k, c(m^j*B, m, k)))
s(n, m, k) = sum(j=0, m^n - 1, c(j, m, k))
test(n, m, k) = s(n, m, k)==a(n+1, m, k)
test1(n, m, k) = s(n, m, k)==(a(n+1, m, k) + if(k<-1, (n<(abs(k)-1)) + sum(i=2, abs(k), (i-1)*b(n+1, m, k, i))))

Is there a way to prove it? Is there another closed form such that we no need to specify cases for $k<-1$?

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