A027058 a(n) = A027052(n, 2n-2).

1, 1, 3, 9, 23, 59, 153, 401, 1063, 2847, 7693, 20947, 57413, 158265, 438467, 1220145, 3408759, 9556815, 26878861, 75815839, 214411865, 607827693, 1726911631, 4916352891, 14022750725, 40066540277, 114666463855, 328662240617

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..750

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( T(n,2*n-2), n=1..30); # 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[T[n,2*n-2], {n,30}] (* 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))
[T(n,2*n-2) for n in (1..30)] # G. C. Greubel, Nov 06 2019

Formula

Conjecture: D-finite with recurrence n*a(n) +(-7*n+4)*a(n-1) +(13*n-10)*a(n-2) +(n-34)*a(n-3) +(-13*n+84)*a(n-4) +(3*n-32)*a(n-5) +(-n+6)*a(n-6) +3*(n-6)*a(n-7)=0. - R. J. Mathar, Jun 15 2020

a(n) ~ 3^(n + 3/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 08 2023

Extensions

Offset changed to 1 and a(1)=1 prepended to sequence by G. C. Greubel, Nov 06 2019