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A055837 T(2n+3,n), where T is the array in A055830.

%I A055837 #15 Jan 22 2020 09:42:33
%S A055837 3,15,73,361,1806,9122,46425,237721,1223365,6321965,32784830,
%T A055837 170528190,889291352,4648068192,24342384337,127707864849,671047979300,
%U A055837 3531026714720,18603737992455,98129545962855,518149580437560
%N A055837 T(2n+3,n), where T is the array in A055830.
%H A055837 G. C. Greubel, <a href="/A055837/b055837.txt">Table of n, a(n) for n = 0..500</a>
%F A055837 Conjecture: 5*n*(n+2)*(11*n-4)*a(n) +(-242*n^3-330*n^2+29*n+42)*a(n-1) -3*(3*n-1)*(11*n+7)*(3*n-2)*a(n-2)=0. - _R. J. Mathar_, Mar 13 2016
%p A055837 with(combinat);
%p A055837 T:= proc(n, k) option remember;
%p A055837       if k<0 or k>n then 0
%p A055837     elif k=0 then fibonacci(n+1)
%p A055837     elif n=1 and k=1 then 0
%p A055837     else T(n-1, k-1) + T(n-1, k) + T(n-2, k)
%p A055837       fi; end:
%p A055837 seq(T(2*n+3, n), n=0..30); # _G. C. Greubel_, Jan 21 2020
%t A055837 T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, Fibonacci[n+1], If[n==1 && k==1, 0, T[n-1, k-1] + T[n-1, k] + T[n-2, k]]]]; Table[T[2*n+3, n], {n,0,30}] (* _G. C. Greubel_, Jan 21 2020 *)
%o A055837 (Sage)
%o A055837 @CachedFunction
%o A055837 def T(n, k):
%o A055837     if (k<0 or k>n): return 0
%o A055837     elif (k==0): return fibonacci(n+1)
%o A055837     elif (n==1 and k==1): return 0
%o A055837     else: return T(n-1, k-1) + T(n-1, k) + T(n-2, k)
%o A055837 [T(2*n+3, n) for n in (0..30)] # _G. C. Greubel_, Jan 21 2020
%Y A055837 Cf. A055830.
%K A055837 nonn
%O A055837 0,1
%A A055837 _Clark Kimberling_, May 28 2000