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A027046 a(n) = Sum_{k=0..n} T(n,k) * T(n,n+k), with T given by A027023.

%I A027046 #15 Nov 05 2019 05:55:31
%S A027046 1,2,5,16,65,286,1305,6232,30377,150178,750937,3785904,19215865,
%T A027046 98086646,503087741,2591128000,13394476993,69466312210,361315132065,
%U A027046 1884242109256,9849626540337,51599350440694,270851460356973
%N A027046 a(n) = Sum_{k=0..n} T(n,k) * T(n,n+k), with T given by A027023.
%H A027046 G. C. Greubel, <a href="/A027046/b027046.txt">Table of n, a(n) for n = 0..1000</a>
%p A027046 T:= proc(n, k) option remember;
%p A027046       if k<3 or k=2*n then 1
%p A027046     else add(T(n-1, k-j), j=1..3)
%p A027046       fi
%p A027046     end:
%p A027046 seq(add(T(n,k)*T(n,n+k), k=0..n), n=0..30); # _G. C. Greubel_, Nov 04 2019
%t A027046 T[n_, k_]:= T[n, k]= If[k<3 || k==2*n, 1, Sum[T[n-1, k-j], {j,3}]]; Table[Sum[T[n,k]*T[n,n+k], {k,0,n}], {n,0,30}] (* _G. C. Greubel_, Nov 04 2019 *)
%o A027046 (Sage)
%o A027046 @CachedFunction
%o A027046 def T(n, k):
%o A027046     if (k<3 or k==2*n): return 1
%o A027046     else: return sum(T(n-1, k-j) for j in (1..3))
%o A027046 [sum(T(n, k)*T(n,n+k) for k in (0..n)) for n in (0..30)] # _G. C. Greubel_, Nov 04 2019
%K A027046 nonn
%O A027046 0,2
%A A027046 _Clark Kimberling_