a(n) = Sum_{k=1..n} sigma(k)*2^(n-k) where sigma(k) = A000203(k) is the sum of divisors of k.

Terms

a(0) =1a(1) =5a(2) =14a(3) =35a(4) =76a(5) =164a(6) =336a(7) =687a(8) =1387a(9) =2792a(10) =5596a(11) =11220a(12) =22454a(13) =44932a(14) =89888a(15) =179807a(16) =359632a(17) =719303a(18) =1438626a(19) =2877294a(20) =5754620a(21) =11509276a(22) =23018576a(23) =46037212a(24) =92074455a(25) =184148952a(26) =368297944a(27) =736595944a(28) =1473191918a(29) =2946383908