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%I #12 Oct 25 2019 04:57:42
%S 1,3,8,20,52,121,301,675,1628,3570,8426,18202,42288,90374,207464,
%T 439800,1000194,2106961,4755715,9967599,22359788,46670273,104154703,
%U 216643945,481381746,998346275,2210037191,4571884119,10088030640
%N a(n) = Sum_{k=0..n} (k+1) * A026736(n,n-k).
%H G. C. Greubel, <a href="/A027220/b027220.txt">Table of n, a(n) for n = 0..1000</a>
%t T[n_, k_]:= T[n, k] = If[k==0 || k==n, 1, If[EvenQ[n] && k==(n-2)/2, T[n-1,k-1] + T[n-2,k-1] + T[n-1,k], T[n-1,k-1] + T[n-1,k]]]; Table[Sum[(k+1)*T[n,n-k], {k, 0, n}], {n, 0, 30}] (* _G. C. Greubel_, Jul 19 2019 *)
%o (Sage)
%o @CachedFunction
%o def T(n, k):
%o if (k==0 or k==n): return 1
%o elif (mod(n, 2)==0 and k==(n-2)/2): return T(n-1, k-1) + T(n-2, k-1)
%o + T(n-1, k)
%o else: return T(n-1, k-1) + T(n-1, k)
%o [sum((k+1)*T(n,n-k) for k in (0..n)) for n in (0..30)] # _G. C. Greubel_, Jul 19 2019
%Y Cf. A026736.
%K nonn
%O 0,2
%A _Clark Kimberling_
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