A027059 a(n) = A027052(n, 2n-3).

0, 2, 6, 18, 52, 146, 406, 1126, 3124, 8684, 24202, 67640, 189576, 532786, 1501254, 4240550, 12005780, 34063896, 96844082, 275848044, 787104288, 2249633916, 6439678858, 18460717684, 52994100984, 152323413890, 438363476086

Offset: 2

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 2..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-3), n=2..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-3], {n,2,30}] (* 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-3) for n in (2..30)] # G. C. Greubel, Nov 06 2019

Formula

Conjecture D-finite with recurrence (n+1)*a(n) +2*(-4*n-1)*a(n-1) +(19*n-5)*a(n-2) +6*(-n-5)*a(n-3) +3*(-7*n+41)*a(n-4) +2*(4*n-29)*a(n-5) +(n+1)*a(n-6) +6*(n-5)*a(n-7)=0. - R. J. Mathar, Jun 15 2020

a(n) ~ 3^(n + 5/2) / (2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 08 2023