• Let $\operatorname{wt}(n)$ be A000120 (i.e., number of $1$'s in binary expansion of $n$).
• Let $a(n,m)$ be the family of integer sequences such that $$ a(n,m) = \sum\limits_{k=1}^{n} [\operatorname{wt}(k) = m]. $$
• Let $T(n,k)$ be A272020 (i.e., irregular triangle read by rows: strictly decreasing sequences of positive numbers given in lexicographic order). Here row $n$ given by the exponents in the binary expansion of $2n$. We also consider that numeration of columns starts with $1$.
• Let $b(n,m)$ be the family of integer sequences such that $$ b(n,m) = \sum\limits_{k=0}^{\min(m,\operatorname{wt}(n+1)-1)} \binom{T(n+1,k+1)-1}{m-k}. $$

I conjecture that $$ b(n,m) = a(n,m). $$

Here is the PARI/GP program to check it numerically:

a(n,m) = sum(k=1, n, hammingweight(k) == m) b(n,m) = my(v1); v1 = Vecrev(binary(n+1)); v1 = Vecrev(select(x->(x>0),v1,1)); sum(k=0, min(m,#v1-1), binomial(v1[k+1]-1,m-k)) test1(n,m) = b(n,m) == a(n,m) 

Is there a way to prove it?

• Let $\operatorname{wt}(n)$ be A000120 (i.e., number of $1$'s in binary expansion of $n$).
• Let $a(n,m)$ be the family of integer sequences such that $$ a(n,m) = \sum\limits_{k=0}^{n} [\operatorname{wt}(k) = m]. $$
• Let $T(n,k)$ be A272020 (i.e., irregular triangle read by rows: strictly decreasing sequences of positive numbers given in lexicographic order). Here row $n$ given by the exponents in the binary expansion of $2n$. We also consider that numeration of columns starts with $1$.
• Let $b(n,m)$ be the family of integer sequences such that $$ b(n,m) = \sum\limits_{k=0}^{\min(m,\operatorname{wt}(n+1)-1)} \binom{T(n+1,k+1)-1}{m-k}. $$

I conjecture that $$ b(n,m) = a(n,m). $$

Here is the PARI/GP program to check it numerically:

a(n,m) = sum(k=1, n, hammingweight(k) == m) b(n,m) = my(v1); v1 = Vecrev(binary(n+1)); v1 = Vecrev(select(x->(x>0),v1,1)); sum(k=0, min(m,#v1-1), binomial(v1[k+1]-1,m-k)) test1(n,m) = b(n,m) == a(n,m) 

Is there a way to prove it?

• Let $\operatorname{wt}(n)$ be A000120 (i.e., number of $1$'s in binary expansion of $n$).
• Let $a(n,m)$ be the family of integer sequences such that $$ a(n,m) = \sum\limits_{k=1}^{n} [\operatorname{wt}(k) = m]. $$
• Let $T(n,k)$ be A272020 (i.e., irregular triangle read by rows: strictly decreasing sequences of positive numbers given in lexicographic order). Here row $n$ given by the exponents in the binary expansion of $2n$. We also consider that numeration of columns start with $1$.
• Let $b(n,m)$ be the family of integer sequences such that $$ b(n,m) = \sum\limits_{k=0}^{\min(m,\operatorname{wt}(n+1)-1)} \binom{T(n+1,k+1)-1}{m-k}. $$

I conjecture that $$ b(n,m) = a(n,m). $$

Here is the PARI/GP program to check it numerically:

a(n,m) = sum(k=1, n, hammingweight(k) == m) b(n,m) = my(v1); v1 = Vecrev(binary(n+1)); v1 = Vecrev(select(x->(x>0),v1,1)); sum(k=0, min(m,#v1-1), binomial(v1[k+1]-1,m-k)) test1(n,m) = b(n,m) == a(n,m) 

Is there a way to prove it?

• Let $\operatorname{wt}(n)$ be A000120 (i.e., number of $1$'s in binary expansion of $n$).
• Let $a(n,m)$ be the family of integer sequences such that $$ a(n,m) = \sum\limits_{k=1}^{n} [\operatorname{wt}(k) = m]. $$
• Let $T(n,k)$ be A272020 (i.e., irregular triangle read by rows: strictly decreasing sequences of positive numbers given in lexicographic order). Here row $n$ given by the exponents in the binary expansion of $2n$. We also consider that numeration of columns starts with $1$.
• Let $b(n,m)$ be the family of integer sequences such that $$ b(n,m) = \sum\limits_{k=0}^{\min(m,\operatorname{wt}(n+1)-1)} \binom{T(n+1,k+1)-1}{m-k}. $$

I conjecture that $$ b(n,m) = a(n,m). $$

Here is the PARI/GP program to check it numerically:

a(n,m) = sum(k=1, n, hammingweight(k) == m) b(n,m) = my(v1); v1 = Vecrev(binary(n+1)); v1 = Vecrev(select(x->(x>0),v1,1)); sum(k=0, min(m,#v1-1), binomial(v1[k+1]-1,m-k)) test1(n,m) = b(n,m) == a(n,m) 

Is there a way to prove it?