A027038 Diagonal sum of right-justified array T given by A027023.

1, 1, 2, 5, 7, 18, 43, 103, 264, 687, 1809, 4836, 13049, 35493, 97218, 267857, 741791, 2063574, 5763595, 16155403, 45429488, 128121191, 362287433, 1026918632, 2917313257, 8304598593, 23685134746, 67669857661, 193652803391

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..750

Crossrefs

Cf. A027023.

Programs

Maple

T:= proc(n, k) option remember;
      if 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,2*n-3*k), k=0..n), n=0..30); # G. C. Greubel, Nov 05 2019

Mathematica

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

Sage

@CachedFunction
def T(n, k):
    if (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, 2*n-3*k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Nov 05 2019

Formula

a(n) = Sum_{k=0..n} T(n-k, 2*n-3*k), where T = A027023. - G. C. Greubel, Nov 05 2019

a(n) ~ 3^(n + 7/2) / (16 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 09 2025