A027069 a(n) = diagonal sum of left-justified array T given by A027052.

1, 1, 1, 2, 2, 4, 5, 7, 11, 14, 22, 32, 43, 67, 97, 134, 206, 298, 419, 637, 923, 1312, 1978, 2872, 4111, 6161, 8961, 12888, 19232, 28010, 40423, 60129, 87665, 126840, 188216, 274634, 398151, 589689, 861001, 1250210, 1848840, 2700900, 3926839, 5799949, 8476579

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,k), k=0..n), n=0..50); # 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[T[n-k,k], {k,0,n}], {n, 0, 50}] (* 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(T(n-k,k) for k in (0..n)) for n in (0..50)] # G. C. Greubel, Nov 06 2019

Formula

a(n) = Sum_{k=0..n} A027052(n - k, k). - Sean A. Irvine, Oct 22 2019

Extensions

More terms from Sean A. Irvine, Oct 22 2019