A026733 a(n) = Sum_{k=0..floor(n/2)} T(n,k), T given by A026725.

1, 1, 3, 5, 13, 23, 57, 103, 249, 455, 1083, 1993, 4693, 8679, 20275, 37633, 87377, 162643, 375789, 701075, 1613413, 3015563, 6916957, 12948083, 29617161, 55513327, 126678893, 237705547, 541325021, 1016736115, 2311294377

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

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

Programs

Maple

A026733 := proc(n)
    add(A026725(n,k),k=0..floor(n/2)) ;
end proc:
seq(A026733(n),n=0..10) ; # R. J. Mathar, Oct 26 2019

Mathematica

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[OddQ[n] && k==(n-1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k]]]; Table[Sum[T[n, k], {k, 0, Floor[n/2]}], {n, 0, 30}] (* G. C. Greubel, Oct 26 2019 *)

PARI

T(n,k) = if(k==n || k==0, 1, if(2*k==n-1, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) ));
vector(31, n, sum(k=0,floor(n-1/2), T(n-1,k)) ) \\ G. C. Greubel, Oct 26 2019

Sage

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==1 and k==(n-1)/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
    else: return T(n-1, k-1) + T(n-1, k)
[sum(T(n, k) for k in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, Oct 26 2019

Formula

Conjecture: (-n+2)*a(n) +(n-2)*a(n-1) +2*(4*n-13)*a(n-2) +8*(-n+4)*a(n-3) +5*(-3*n+14)*a(n-4) +(15*n-94)*a(n-5) +2*(-2*n+9)*a(n-6) +4*(n-6)*a(n-7)=0. - R. J. Mathar, Oct 26 2019