A027268 a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A026536.

0, 6, 20, 180, 644, 5502, 20292, 174456, 654632, 5673140, 21528000, 187675644, 717800628, 6284986554, 24178479500, 212408191568, 820811282352, 7229648901024, 28037230854096, 247468885359240, 962488105227160, 8510025522045036, 33177800527098040, 293772371437293720

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A026536, A027267, A027269.

Programs

Mathematica

T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[n/2], If[EvenQ[n], T[n-1, k-2] +T[n-1, k-1] +T[n-1, k], T[n-1, k-2] +T[n-1, k]] ]];
Table[Sum[T[n, k]*T[n,k+1], {k,0,2*n-1}], {n,40}] (* G. C. Greubel, Apr 12 2022 *)

SageMath

@CachedFunction
def T(n, k): # A026536
    if k < 0 or n < 0: return 0
    elif k == 0 or k == 2*n: return 1
    elif k == 1 or k == 2*n-1: return n//2
    elif n % 2 == 1: return T(n-1, k-2) + T(n-1, k)
    return T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
def A027268(n): return sum(T(n,k)*T(n,k+1) for k in (0..2*n-1))
[A027268(n) for n in (1..40)] # G. C. Greubel, Apr 12 2022

Formula

a(n) = Sum_{k=0..2n-1} A026536(n,k) * A026536(n,k+1)

Extensions

More terms from Sean A. Irvine, Oct 26 2019