Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

Terms

a(0) =1a(1) =2a(2) =2a(3) =3a(4) =5a(5) =5a(6) =15a(7) =8a(8) =8a(9) =35a(10) =33a(11) =13a(12) =80a(13) =131a(14) =48a(15) =21a(16) =171a(17) =409a(18) =279a(19) =34a(20) =355a(21) =1180a(22) =1375a(23) =384a(24) =55a(25) =715a(26) =3128a(27) =5335a(28) =2895a(29) =89