A027040 a(n) = self-convolution of row n of array T given by A027023.

1, 3, 9, 31, 129, 531, 2129, 8351, 32177, 122211, 458801, 1706015, 6293169, 23057651, 83992313, 304424639, 1098525761, 3948727555, 14145206209, 50515602111, 179904080257, 639103899411, 2265253438745, 8012421964063

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 (n<0 or k>2*n) then 0
    elif k<3 or k=2*n then 1
    else add(T(n-1, k-j), j=1..3)
      fi
    end:
seq( add(T(n,k)*T(n,2*n-k), k=0..2*n), n=0..30); # G. C. Greubel, Nov 05 2019

Mathematica

T[n_, k_]:= T[n, k]= If[n<0 || k>2*n, 0, If[k<3 || k==2*n, 1, Sum[T[n-1, k-j], {j, 3}]]]; Table[Sum[T[n, k]*T[n, 2*n-k], {k, 0, 2*n}], {n, 0, 30}] (* G. C. Greubel, Nov 05 2019 *)

Sage

@CachedFunction
def T(n, k):
    if (n<0 or k>2*n): return 0
    elif (k<3 or k==2*n): return 1
    else: return sum(T(n-1, k-j) for j in (1..3))
[sum(T(n,k)*T(n,2*n-k) for k in (0..2*n)) for n in (4..30)] # G. C. Greubel, Nov 05 2019

Formula

a(n) = Sum_{k=0..2*n} T(n,k)*T(n,2*n-k), where T = A027023. - G. C. Greubel, Nov 05 2019