A027263 a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A026519.

2, 6, 52, 180, 1516, 5502, 46936, 174456, 1504432, 5673140, 49288856, 187675644, 1639174304, 6284986554, 55108565584, 212408191568, 1868067054968, 7229648901024, 63734526307552, 247468885359240, 2185849699156352, 8510025522045036, 75288454939134992, 293772371437293720

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026532, A026533, A026534, A027262, A027264, A027265, A027266.

Programs

Mathematica

T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+1)/2], If[EvenQ[n], T[n-1, k-2] + T[n-1, k], T[n-1, k-1] + T[n-1, k-2] + T[n-1, k] ]]]]; (* T = A026519 *)
a[n_]:= a[n]= Block[{$RecursionLimit = Infinity}, Sum[T[n, k]*T[n, k+1], {k, 0, 2*n-1}] ];
Table[a[n], {n, 40}] (* G. C. Greubel, Dec 21 2021 *)

Sage

@CachedFunction
def T(n,k): # T = A026519
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+1)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
@CachedFunction
def a(n): return sum( T(n,k)*T(n,k+1) for k in (0..2*n-1) )
[a(n) for n in (1..40)] # G. C. Greubel, Dec 21 2021

Formula

a(n) = Sum_{k=0..2n-1} A026519(n,k) * A026519(n,k+1).

Extensions

More terms from Sean A. Irvine, Oct 26 2019