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  • Let $a(n)$ be A290383 i.e. number of set partitions of $[n]$ such that the smallest element of each block is odd. Here $$ a(n)=b(n,0,0) $$ where $$ b(n,m,t)=\sum\limits_{j=1}^{m-t+1}b(n-1,\max(m,j),1-t), \\ b(0,m,t)=1 $$
  • Let $$ R_1(n,q)=(q+1)^2R_1(n-1,q)+(q+1)R_1(n-1,q+1), \\ R_1(0,q)=1 $$
  • Let $$ R_2(n,q)=(q+1)^2R_2(n-1,q)+(q+2)R_2(n-1,q+1), \\ R_2(0,q)=1 $$

I conjecture that $$ R_1(n,0)=a(2n+1), \\ R_2(n,0)=a(2(n+1)). $$

Is there a way to prove it?

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