A026623 a(n) = Sum_{k=0..floor(n/2)} A026615(n, k).

1, 1, 4, 6, 18, 27, 72, 111, 283, 447, 1112, 1791, 4381, 7167, 17305, 28671, 68497, 114687, 271560, 458751, 1077949, 1835007, 4283069, 7340031, 17031503, 29360127, 67768777, 117440511, 269797323, 469762047, 1074583315, 1879048191

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

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

Crossrefs

Cf. A026615, A026616, A026617, A026618, A026619, A026620, A026621.

Cf. A026622, A026624, A026625, A026956, A026957, A026958, A026959.

Cf. A026960.

Programs

Magma

I:=[1,4,6]; [1] cat [n le 3 select I[n] else ( 2*(7*n-24)*(7*n-29)*(n-1)*Self(n-1) + 4*(7*n-8)*(7*n-31)*(n-3)*Self(n-2) - 8*(7*n-15)*(7*n-24)*(n-4)*Self(n-3) - (147*n^3 - 1603*n^2 + 5896*n - 7152))/(n*(7*n-31)*(7*n-22)): n in [1..41]]; // G. C. Greubel, Jun 15 2024

Mathematica

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k==1 || k==n-1, 2*n-1, T[n -1, k-1] + T[n-1,k]]]; (* T = A026615 *)
A026623[n_]:= Sum[T[n, k], {k, 0, Floor[n/2]}];
Table[A026623[n], {n,0,40}] (* G. C. Greubel, Jun 15 2024 *)

SageMath

@CachedFunction
def T(n, k): # T = A026615
    if k==0 or k==n: return 1
    elif k==1 or k==n-1: return 2*n-1
    else: return T(n-1, k-1) + T(n-1, k)
def A026623(n): return sum(T(n,k) for k in range((n//2)+1))
[A026623(n) for n in range(41)] # G. C. Greubel, Jun 15 2024

Formula

a(n) = ( 2*(7*n-24)*(7*n-29)*(n-1)*a(n-1) + 4*(7*n-8)*(7*n-31)*(n-3)*a(n-2) - 8*(7*n-15)*(7*n-24)*(n-4)*a(n-3) - (147*n^3 - 1603*n^2 + 5896*n - 7152))/(n*(7*n-31)*(7*n-22)), for n > 3, with a(0) = a(1) = 1, a(2) = 4, and a(3) = 6. - G. C. Greubel, Jun 15 2024