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

1,2

Links

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

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A026736.

Sequence in context: A352815 A007167 A036728 * A124541 A353264 A117669

Adjacent sequences: A027213 A027214 A027215 * A027217 A027218 A027219

Keyword

nonn

Author

Clark Kimberling

Status

approved