A027075 a(n) = Sum_{k=0..2n} (k+1) * A027052(n, k).

1, 4, 17, 58, 199, 682, 2301, 7654, 25145, 81740, 263407, 842720, 2679935, 8479378, 26713555, 83847748, 262335577, 818473148, 2547289679, 7910433568, 24517303535, 75854736178, 234317624167, 722776320072, 2226565995913

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Programs

Maple

T:= proc(n, k) option remember;
      if k<0 or k>2*n then 0
    elif k=0 or k=2 or k=2*n then 1
    elif k=1 then 0
    else add(T(n-1, k-j), j=1..3)
      fi
    end:
seq( add((k+1)*T(n,k), k=0..2*n), n=0..30); # G. C. Greubel, Nov 06 2019

Mathematica

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 *)

Sage

@CachedFunction
def T(n, k):
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2 or k==2*n): return 1
    elif (k==1): return 0
    else: return sum(T(n-1, k-j) for j in (1..3))
[sum((k+1)*T(n,k) for k in (0..2*n)) for n in (0..30)] # G. C. Greubel, Nov 06 2019