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A027044 a(n) = Sum_{k=0..2n} (k+1) * A027023(n,2n-k).

%I A027044 #13 Nov 05 2019 01:01:03
%S A027044 1,6,19,56,165,486,1435,4248,12601,37438,111367,331608,988181,2946662,
%T A027044 8791447,26241632,78359825,234069830,699404127,2090385216,6249236653,
%U A027044 18686125070,55884824535,167164064984,500102988889
%N A027044 a(n) = Sum_{k=0..2n} (k+1) * A027023(n,2n-k).
%H A027044 G. C. Greubel, <a href="/A027044/b027044.txt">Table of n, a(n) for n = 0..1000</a>
%p A027044 T:= proc(n, k) option remember;
%p A027044       if k<3 or k=2*n then 1
%p A027044     else add(T(n-1, k-j), j=1..3)
%p A027044       fi
%p A027044     end:
%p A027044 seq(add((k+1)*T(n,2*n-k), k=0..2*n), n=0..30); # _G. C. Greubel_, Nov 04 2019
%t A027044 T[n_, k_]:= T[n, k]= If[k<3 || k==2*n, 1, Sum[T[n-1, k-j], {j,3}]]; Table[Sum[(k+1)*T[n,2*n-k], {k,0,2*n}], {n,0,30}] (* _G. C. Greubel_, Nov 04 2019 *)
%o A027044 (Sage)
%o A027044 @CachedFunction
%o A027044 def T(n, k):
%o A027044     if (k<3 or k==2*n): return 1
%o A027044     else: return sum(T(n-1, k-j) for j in (1..3))
%o A027044 [sum((k+1)*T(n, 2*n-k) for k in (0..2*n)) for n in (0..30)] # _G. C. Greubel_, Nov 04 2019
%K A027044 nonn
%O A027044 0,2
%A A027044 _Clark Kimberling_