A027986 a(n) = Sum_{k=0..2*n-2} T(n, k)*T(n, k+2), T given by A027960.

20, 100, 420, 1694, 6746, 26735, 105722, 417613, 1648692, 6507197, 25681362, 101359219, 400094756, 1579568360, 6237401648, 24635828774, 97327071806, 384596629610, 1520138671688, 6009885957464, 23765835521966, 94003008751940

Offset: 2

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 2..1000

Crossrefs

Cf. A027960.

Programs

Magma

f:= func< n,k | (&+[Binomial(2*n-k+j,j)*Lucas(2*(k-n-j)): j in [0..k-n-1]]) >;
A027960:= func< n,k | k le n select Lucas(k+1) else Lucas(k+1) - f(n,k) >;
A027986:= func< n | (&+[A027960(n,k)*A027960(n,k+2): k in [0..2*n-2]]) >;
[A027986(n): n in [2..40]]; // G. C. Greubel, Jun 14 2025

Mathematica

f[n_, k_]:= f[n,k]= Sum[Binomial[2*n-k+j,j]*LucasL[2*(k-n-j)], {j,0,k-n-1}];
A027960[n_, k_]:= LucasL[k+1] -f[n,k]*Boole[k>n];
A027986[n_]:= A027986[n]= Sum[A027960[n,k]*A027960[n,k+2], {k,0,2*n-2}];
Table[A027986[n], {n,2,40}] (* G. C. Greubel, Jun 14 2025 *)

SageMath

@CachedFunction
def T(n, k): # T = A027960
    if (k>2*n): return 0
    elif (kA027986(n): return sum(T(n,k)*T(n,k+2) for k in range(2*n-1))
print([A027986(n) for n in range(2,41)]) # G. C. Greubel, Jun 14 2025

Formula

a(n) = A000032(2*n+1) - 7 + 3*(n mod 2) + Sum_{k=n-1..2*n-2} A027960(n, k)*A027960(n, k+2). - G. C. Greubel, Jun 14 2025