%I #13 Nov 07 2019 08:28:38

%S 1,4,17,58,199,682,2301,7654,25145,81740,263407,842720,2679935,

%T 8479378,26713555,83847748,262335577,818473148,2547289679,7910433568,

%U 24517303535,75854736178,234317624167,722776320072,2226565995913

%N a(n) = Sum_{k=0..2n} (k+1) * A027052(n, k).

%H G. C. Greubel, <a href="/A027075/b027075.txt">Table of n, a(n) for n = 0..1000</a>

%p T:= proc(n, k) option remember;

%p if k<0 or k>2*n then 0

%p elif k=0 or k=2 or k=2*n then 1

%p elif k=1 then 0

%p else add(T(n-1, k-j), j=1..3)

%p fi

%p end:

%p seq( add((k+1)*T(n,k), k=0..2*n), n=0..30); # _G. C. Greubel_, Nov 06 2019

%t T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j, 3}]]]]; Table[Sum[(k+1)*T[n,k], {k, 0, 2*n}], {n,0,30}] (* _G. C. Greubel_, Nov 06 2019 *)

%o (Sage)

%o @CachedFunction

%o def T(n, k):

%o if (k<0 or k>2*n): return 0

%o elif (k==0 or k==2 or k==2*n): return 1

%o elif (k==1): return 0

%o else: return sum(T(n-1, k-j) for j in (1..3))

%o [sum((k+1)*T(n,k) for k in (0..2*n)) for n in (0..30)] # _G. C. Greubel_, Nov 06 2019

%K nonn

%O 0,2

%A _Clark Kimberling_