The Faulhaber-Knuth a(0,n) sequence.

Terms

a(0) =2a(1) =1a(2) =1a(3) =1a(4) =1a(5) =0a(6) =0a(7) =1a(8) =37a(9) =-60a(10) =-5a(11) =37a(12) =174a(13) =-955a(14) =-10545a(15) =38610a(16) =176297a(17) =-322740a(18) =-205420a(19) =4512655a(20) =56820585a(21) =-104019264a(22) =-25907081a(23) =94854194a(24) =1141847218a(25) =-2090335775a(26) =-414239903275