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A055838 T(2n+4,n), where T is the array in A055830.

%I A055838 #12 Jan 21 2020 10:08:38
%S A055838 5,30,162,850,4425,22995,119560,622512,3246750,16963375,88779900,
%T A055838 465386220,2443204946,12844119225,67608235800,356288599640,
%U A055838 1879625199825,9925931817045,52464942758250,277546278287250
%N A055838 T(2n+4,n), where T is the array in A055830.
%H A055838 G. C. Greubel, <a href="/A055838/b055838.txt">Table of n, a(n) for n = 0..500</a>
%F A055838 Conjecture: 5*n*(n+3)*(n-1)*a(n) -2*(n-1)*(11*n+8)*(n+2)*a(n-1) -3*(3*n-1)*(3*n-2)*(n+1)*a(n-2)=0. - _R. J. Mathar_, Mar 13 2016
%p A055838 with(combinat);
%p A055838 T:= proc(n, k) option remember;
%p A055838       if k<0 or k>n then 0
%p A055838     elif k=0 then fibonacci(n+1)
%p A055838     elif n=1 and k=1 then 0
%p A055838     else T(n-1, k-1) + T(n-1, k) + T(n-2, k)
%p A055838       fi; end:
%p A055838 seq(T(2*n+4, n), n=0..30); # _G. C. Greubel_, Jan 21 2020
%t A055838 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+4, n], {n,0,30}] (* _G. C. Greubel_, Jan 21 2020 *)
%o A055838 (Sage)
%o A055838 @CachedFunction
%o A055838 def T(n, k):
%o A055838     if (k<0 and k>n): return 0
%o A055838     elif (k==0): return fibonacci(n+1)
%o A055838     elif (n==1 and k==1): return 0
%o A055838     else: return T(n-1, k-1) + T(n-1, k) + T(n-2, k)
%o A055838 [T(2*n+4, n) for n in (0..30)] # _G. C. Greubel_, Jan 21 2020
%Y A055838 Cf. A055830.
%K A055838 nonn
%O A055838 0,1
%A A055838 _Clark Kimberling_, May 28 2000