A027216 a(n) = Sum_{k=0..n-1} T(n,k)*T(n,k+1), T given by A026736.

1, 4, 15, 63, 237, 1034, 3945, 17577, 67640, 304902, 1179415, 5352038, 20771331, 94628132, 368083879, 1680820301, 6548692260, 29946087674, 116816782997, 534628747310

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

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

Crossrefs

Cf. A026736.

Programs

GAP

T:= function(n, k)
    if k=0 or k=n then return 1;
    elif k=n-1 then return n;
    elif (n mod 2)=0 and k=Int((n-2)/2) then return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k);
    else return T(n-1, k-1) + T(n-1, k);
    fi;
  end;
List([1..20], n-> Sum([0..n-1], k-> T(n, k)*T(n,k+1) )); # G. C. Greubel, Jul 19 2019

Mathematica

T[n_, k_]:= T[n, k] = If[k==0 || k==n, 1, If[EvenQ[n] && k==(n-2)/2, T[n-1,k-1] + T[n-2,k-1] + T[n-1,k], T[n-1,k-1] + T[n-1,k]]]; Table[Sum[T[n,k]*T[n,k+1], {k, 0, n-1}], {n, 1, 30}] (* G. C. Greubel, Jul 19 2019 *)

PARI

T(n, k) = if(k==n || k==0, 1, k==n-1, n, if((n%2)==0 && k==(n-2)/2, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) ));
vector(20, n, sum(k=0, n-1, T(n, k)*T(n,k+1)) ) \\ G. C. Greubel, Jul 19 2019

Sage

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n, 2)==0 and k==(n-2)/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
    else: return T(n-1, k-1) + T(n-1, k)
[sum(T(n,k)*T(n,k+1) for k in (0..n-1)) for n in (1..30)] # G. C. Greubel, Jul 19 2019

Formula

a(n) ~ (1/2 - (-1)^n/10) * phi^(3*n - 5/2 + (-1)^n/2), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 19 2019