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A027058 a(n) = A027052(n, 2n-2).

%I A027058 #16 Mar 08 2023 04:17:04
%S A027058 1,1,3,9,23,59,153,401,1063,2847,7693,20947,57413,158265,438467,
%T A027058 1220145,3408759,9556815,26878861,75815839,214411865,607827693,
%U A027058 1726911631,4916352891,14022750725,40066540277,114666463855,328662240617
%N A027058 a(n) = A027052(n, 2n-2).
%H A027058 G. C. Greubel, <a href="/A027058/b027058.txt">Table of n, a(n) for n = 1..750</a>
%F A027058 Conjecture: D-finite with recurrence n*a(n) +(-7*n+4)*a(n-1) +(13*n-10)*a(n-2) +(n-34)*a(n-3) +(-13*n+84)*a(n-4) +(3*n-32)*a(n-5) +(-n+6)*a(n-6) +3*(n-6)*a(n-7)=0. - _R. J. Mathar_, Jun 15 2020
%F A027058 a(n) ~ 3^(n + 3/2) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Mar 08 2023
%p A027058 T:= proc(n, k) option remember;
%p A027058       if k<0 or k>2*n then 0
%p A027058     elif k=0 or k=2 or k=2*n then 1
%p A027058     elif k=1 then 0
%p A027058     else add(T(n-1, k-j), j=1..3)
%p A027058       fi
%p A027058     end:
%p A027058 seq( T(n,2*n-2), n=1..30); # _G. C. Greubel_, Nov 06 2019
%t A027058 T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j, 3}]]]]; Table[T[n,2*n-2], {n,30}] (* _G. C. Greubel_, Nov 06 2019 *)
%o A027058 (Sage)
%o A027058 @CachedFunction
%o A027058 def T(n, k):
%o A027058     if (k<0 or k>2*n): return 0
%o A027058     elif (k==0 or k==2 or k==2*n): return 1
%o A027058     elif (k==1): return 0
%o A027058     else: return sum(T(n-1, k-j) for j in (1..3))
%o A027058 [T(n,2*n-2) for n in (1..30)] # _G. C. Greubel_, Nov 06 2019
%K A027058 nonn
%O A027058 1,3
%A A027058 _Clark Kimberling_
%E A027058 Offset changed to 1 and a(1)=1 prepended to sequence by _G. C. Greubel_, Nov 06 2019