A027078 a(n) = Sum_{k=0..n} T(n,k) * T(n,n+k), with T given by A027052.

1, 0, 2, 8, 31, 130, 590, 2798, 13541, 66724, 332708, 1673536, 8479367, 43218034, 221383712, 1138976166, 5882112985, 30479772624, 158413903096, 825556260636, 4312814257059, 22580855859166, 118468635595680, 622698941708890

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(T(n,k)*T(n,n+k), k=0..n), n=0..30); # G. C. Greubel, Nov 07 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[T[n, k]*T[n, n+k], {k, 0, n}], {n,30}] (* G. C. Greubel, Nov 07 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(T(n,k)*T(n,n+k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Nov 07 2019