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A352972 a(n) = Sum_{j=0..2*n} Sum_{k=0..j} A026536(j, k).

%I A352972 #12 Apr 14 2022 01:21:00
%S A352972 1,6,35,204,1199,7089,42070,250269,1491262,8896310,53118352,317373194,
%T A352972 1897253203,11346582851,67882263130,406231442387,2431626954934,
%U A352972 14558306758418,87177151134954,522110098886882,3127380060424476,18734897945679836,112245303177542790,672552484035697364,4030148584900522009
%N A352972 a(n) = Sum_{j=0..2*n} Sum_{k=0..j} A026536(j, k).
%H A352972 G. C. Greubel, <a href="/A352972/b352972.txt">Table of n, a(n) for n = 0..500</a>
%F A352972 a(n) = Sum_{j=0..2*n} Sum_{k=0..j} A026536(j, k).
%t A352972 T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[n/2], If[EvenQ[n], T[n-1, k-2] +T[n-1, k-1] +T[n-1, k], T[n-1, k-2] +T[n-1, k]] ]];
%t A352972 A352972[n_]:= A352972[n]= Sum[T[j,k], {j,0,2*n}, {k,0,j}];
%t A352972 Table[A352972[n], {n,0,40}]
%o A352972 (SageMath)
%o A352972 @CachedFunction
%o A352972 def T(n, k): # A026536
%o A352972     if k == 0 or k == 2*n: return 1
%o A352972     elif k == 1 or k == 2*n-1: return n//2
%o A352972     elif n % 2 == 1: return T(n-1, k-2) + T(n-1, k)
%o A352972     return T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
%o A352972 def A352972(n): return sum(sum(T(j,k) for k in (0..j)) for j in (0..2*n))
%o A352972 [A352972(n) for n in (3..40)]
%Y A352972 Cf. A026536, A026550.
%K A352972 nonn
%O A352972 0,2
%A A352972 _G. C. Greubel_, Apr 12 2022