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We have general recurrence for A243499 (which is product of parts of integer partitions as enumerated in the table A125106) $$a(n)=(1+b(n))a(t(n)), a(0)=1$$ where $b(n)$ is A023416 (which is number of …

Let $$g(n,m)=\sum\limits_{i=0}^{n}\sum\limits_{j=0}^{m}\binom{i+j}{j}\binom{n-i+m-j}{m-j}\alpha^i\beta^j$$ I conjecture that $$g(n,m)=\binom{n+m}{m}f(n,m)$$ Here is the PARI prog to verify this conjec …

Let $m\geq 2$ be a fixed integer. Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$. …

Let $\operatorname{wt}(n)$ be A000120, i.e. the number of $1$'s in binary expansion of $n$ (or the binary weight of $n$). Let $a(n)$ be the sequence of numbers $k$ such that $\operatorname{wt}(k)\oper …

We have a sequence which generalize A329369: $$a(2n+1, p, q) = a(n, p,q), a(2n, p , q) = pa(n, p,q) + qa(n - 2^{f(n)}, p,q) + a(2n - 2^{f(n)}, p,q), a(0, p, q) = 1$$ where $f(n)$ is A007814, exponent …

Let $m\geq 2$ be a fixed integer. Let $$f(n):=\begin{cases} mf\left(\frac{n}{m}\right),&\text{if $n\mod m = 0$;}\\ 1,&\text{otherwise} \end{cases}$$ then if we have $$a(n):=\begin{cases} 1,&\text{if $ …

Let $m\geqslant1$ be a fixed integer. Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of …

Let $p$ and $q$ be integers. Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$. Then …

Let $f(n)$ be A053645, distance to largest power of $2$ less than or equal to $n$; write $n$ in binary, change the first digit to zero, and convert back to decimal. Let $g(n)$ be A007814, the expone …

Given $n$ balls, all of which are initially in the first of $n$ numbered boxes, $a(n)$ is the number of steps required to get one ball in each box when a step consists of moving to the next box every  …

Let $a(n)$ be A290254, the viabin numbers of the self-conjugate integer partitions, also defined as $\left\lbrace 0 \right\rbrace$ union fixed points of A059894, self-inverse permutation defined as fo …

Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$. Then we have an integer sequence …

Let $m \geqslant 1$ be a fixed integer. Let $\operatorname{wt}(n)$ be A000120, $1$'s-counting sequence: number of $1$'s in binary expansion of $n$ (or the binary weight of $n$). Let $f(n)$ be A007814, …

Let $m \geqslant 1$ be a fixed integer. Let $f(n)$ be A007814, exponent of highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$. Th …

Related questions: Number of open tours by a biased rook on a specific $f(n)\times 1$ board which end on a $k$-th cell from the right Sum with products turned into subsequences Combinatorial interpre …