A027276 a(n) = Sum_{k=0..2n} (k+1) * A026552(n, k).

1, 6, 27, 72, 270, 648, 2268, 5184, 17496, 38880, 128304, 279936, 909792, 1959552, 6298560, 13436928, 42830208, 90699264, 287214336, 604661760, 1904684544, 3990767616, 12516498432, 26121388032, 81629337600, 169789022208

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,12,0,-36).

Crossrefs

Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026564, A026566, A026567, A027272, A027273, A027274, A027275.

Programs

Magma

I:= [6,27,72,270]; [1] cat [n le 4 select I[n] else 12*(Self(n-2) - 3*Self(n-4)): n in [1..41]]; // G. C. Greubel, Dec 18 2021

Mathematica

Table[-(1/2)*Boole[n==0] + (1/4)*6^(n/2)*(n+1)*(3*(1+(-1)^n) + Sqrt[6]*(1-(-1)^n)), {n, 0, 40}] (* G. C. Greubel, Dec 18 2021 *)

PARI

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -36,0,12,0]^n*[1;6;27;72])[1,1] \\ Charles R Greathouse IV, Oct 21 2022

Sage

@CachedFunction
def T(n,k): # T = A026552
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+2)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-2)
@CachedFunction
def a(n): return sum( (k+1)*T(n,k) for k in (0..2*n) )
[a(n) for n in (0..40)] # G. C. Greubel, Dec 18 2021

Formula

a(n) = Sum_{k=0..2n} (k+1) * A026552(n, k).

G.f.: (1 +6*x +15*x^2 -18*x^3)/(1-6*x^2)^2.

a(n) = -(1/2)*[n=0] + (1/4)*6^(n/2)*(n + 1)*(3*(1 + (-1)^n) + sqrt(6)*(1 - (-1)^n)). - G. C. Greubel, Dec 18 2021