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%I #15 Jul 19 2019 20:24:03
%S 1,4,15,63,237,1034,3945,17577,67640,304902,1179415,5352038,20771331,
%T 94628132,368083879,1680820301,6548692260,29946087674,116816782997,
%U 534628747310
%N a(n) = Sum_{k=0..n-1} T(n,k)*T(n,k+1), T given by A026736.
%H G. C. Greubel, <a href="/A027216/b027216.txt">Table of n, a(n) for n = 1..1000</a>
%F 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 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 (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 vector(20, n, sum(k=0, n-1, T(n, k)*T(n,k+1)) ) \\ _G. C. Greubel_, Jul 19 2019
%o (Sage)
%o @CachedFunction
%o def T(n, k):
%o if (k==0 or k==n): return 1
%o elif (mod(n, 2)==0 and k==(n-2)/2): return T(n-1, k-1) + T(n-2, k-1)
%o + T(n-1, k)
%o else: return T(n-1, k-1) + T(n-1, k)
%o [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 (GAP)
%o T:= function(n, k)
%o if k=0 or k=n then return 1;
%o elif k=n-1 then return n;
%o 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 else return T(n-1, k-1) + T(n-1, k);
%o fi;
%o end;
%o List([1..20], n-> Sum([0..n-1], k-> T(n, k)*T(n,k+1) )); # _G. C. Greubel_, Jul 19 2019
%Y Cf. A026736.
%K nonn
%O 1,2
%A _Clark Kimberling_
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