A sum over partitions (q=6), see first comment.

Terms

a(0) =1a(1) =5a(2) =35a(3) =210a(4) =1290a(5) =7735a(6) =46620a(7) =279685a(8) =1679370a(9) =10076190a(10) =60464670a(11) =362787810a(12) =2176773305a(13) =13060638360a(14) =78364108620a(15) =470184650495