A027044 a(n) = Sum{(k+1)*T(n,2n-k)}, 0<=k<=2n, T given by A027023.

1, 6, 19, 56, 165, 486, 1435, 4248, 12601, 37438, 111367, 331608, 988181, 2946662, 8791447, 26241632, 78359825, 234069830, 699404127, 2090385216, 6249236653, 18686125070, 55884824535, 167164064984, 500102988889