Let $k_1,k_2,\cdots,k_n,\cdots$ be a sequence of known positive numbers. Define $$ a_1:=k_1,\\ a_2:=C_2^2k_2+C_2^1k_1a_1,\\ a_3:=C_3^3k_3+C_3^2k_2a_1+C_3^1k_1a_2,\\ a_4:=C_4^4k_4+C_4^3k_3a_1+C_4^2k_2a_2+C_4^1k_1a_3,\\ \cdots\\ a_n:=C_n^nk_n+C_n^{n1}k_{n1}a_1+C_n^{n2}k_{n2}a_2+\cdots+C_n^1k_1a_{n1}. $$ What is the general term formula for $a_n$?

4$\begingroup$ Let $A(x) = \sum_{n=0}^\infty a_n x^n/n!$, where $a_0=1$, and let $K(x) = \sum_{n=1}^\infty k_n x^n/n!$. Then $A(x) = 1/(1K(x))$. $\endgroup$ – Ira Gessel May 15 '19 at 21:33

$\begingroup$ Could you please provide more details? I still can not get the general formula for $a_n$. Thank you very much! $\endgroup$ – Wenguang Zhao May 16 '19 at 4:22

$\begingroup$ Expand $1/(1K(x))= \sum_n K(x)^n$ by the multinomial theorem. See my answer to mathoverflow.net/questions/53384/…. $\endgroup$ – Ira Gessel May 16 '19 at 5:10

$\begingroup$ Thanks very much! $\endgroup$ – Wenguang Zhao May 16 '19 at 6:21