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A027059 a(n) = A027052(n, 2n-3).

%I A027059 #16 Mar 08 2023 04:19:08
%S A027059 0,2,6,18,52,146,406,1126,3124,8684,24202,67640,189576,532786,1501254,
%T A027059 4240550,12005780,34063896,96844082,275848044,787104288,2249633916,
%U A027059 6439678858,18460717684,52994100984,152323413890,438363476086
%N A027059 a(n) = A027052(n, 2n-3).
%H A027059 G. C. Greubel, <a href="/A027059/b027059.txt">Table of n, a(n) for n = 2..750</a>
%F A027059 Conjecture D-finite with recurrence (n+1)*a(n) +2*(-4*n-1)*a(n-1) +(19*n-5)*a(n-2) +6*(-n-5)*a(n-3) +3*(-7*n+41)*a(n-4) +2*(4*n-29)*a(n-5) +(n+1)*a(n-6) +6*(n-5)*a(n-7)=0. - _R. J. Mathar_, Jun 15 2020
%F A027059 a(n) ~ 3^(n + 5/2) / (2 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Mar 08 2023
%p A027059 T:= proc(n, k) option remember;
%p A027059       if k<0 or k>2*n then 0
%p A027059     elif k=0 or k=2 or k=2*n then 1
%p A027059     elif k=1 then 0
%p A027059     else add(T(n-1, k-j), j=1..3)
%p A027059       fi
%p A027059     end:
%p A027059 seq( T(n,2*n-3), n=2..30); # _G. C. Greubel_, Nov 06 2019
%t A027059 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-3], {n,2,30}] (* _G. C. Greubel_, Nov 06 2019 *)
%o A027059 (Sage)
%o A027059 @CachedFunction
%o A027059 def T(n, k):
%o A027059     if (k<0 or k>2*n): return 0
%o A027059     elif (k==0 or k==2 or k==2*n): return 1
%o A027059     elif (k==1): return 0
%o A027059     else: return sum(T(n-1, k-j) for j in (1..3))
%o A027059 [T(n,2*n-3) for n in (2..30)] # _G. C. Greubel_, Nov 06 2019
%K A027059 nonn
%O A027059 2,2
%A A027059 _Clark Kimberling_