cp's OEIS Frontend

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

%I A027216 #19 Sep 29 2024 17:00:19
%S A027216 1,4,15,63,237,1034,3945,17577,67640,304902,1179415,5352038,20771331,
%T A027216 94628132,368083879,1680820301,6548692260,29946087674,116816782997,
%U A027216 534628747310
%N A027216 a(n) = Sum_{k=0..n-1} T(n,k)*T(n,k+1), T given by A026736.
%H A027216 G. C. Greubel, <a href="/A027216/b027216.txt">Table of n, a(n) for n = 1..1000</a>
%F A027216 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
%t A027216 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 *)
%o A027216 (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) ));
%o A027216 vector(20, n, sum(k=0, n-1, T(n, k)*T(n,k+1)) ) \\ _G. C. Greubel_, Jul 19 2019
%o A027216 (Sage)
%o A027216 @CachedFunction
%o A027216 def T(n, k):
%o A027216     if (k==0 or k==n): return 1
%o A027216     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)
%o A027216     else: return T(n-1, k-1) + T(n-1, k)
%o A027216 [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
%o A027216 (GAP)
%o A027216 T:= function(n, k)
%o A027216     if k=0 or k=n then return 1;
%o A027216     elif k=n-1 then return n;
%o A027216     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);
%o A027216     else return T(n-1, k-1) + T(n-1, k);
%o A027216     fi;
%o A027216   end;
%o A027216 List([1..20], n-> Sum([0..n-1], k-> T(n, k)*T(n,k+1) )); # _G. C. Greubel_, Jul 19 2019
%Y A027216 Cf. A026736.
%K A027216 nonn
%O A027216 1,2
%A A027216 _Clark Kimberling_