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A027219 a(n) = Sum_{k=0..n} (k+1) * A026736(n,k).

%I A027219 #16 Sep 29 2024 17:01:02
%S A027219 1,3,8,20,50,117,283,639,1512,3338,7774,16898,38884,83566,190488,
%T A027219 405848,918120,1942813,4367665,9191499,20555546,43061789,95874233,
%U A027219 200083005,443770612,923124007,2040635445,4233080627,9330343290
%N A027219 a(n) = Sum_{k=0..n} (k+1) * A026736(n,k).
%H A027219 G. C. Greubel, <a href="/A027219/b027219.txt">Table of n, a(n) for n = 0..1000</a>
%t A027219 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,k], {k, 0, n}], {n, 0, 30}] (* _G. C. Greubel_, Jul 19 2019 *)
%o A027219 (PARI) T(n, k) = if(k==n || k==0, 1, k==n-1, n, if((n%2)==0 && 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) ));
%o A027219 vector(20, n, n--; sum(k=0, n, (k+1)*T(n, k)) ) \\ _G. C. Greubel_, Jul 19 2019
%o A027219 (Sage)
%o A027219 @CachedFunction
%o A027219 def T(n, k):
%o A027219     if (k==0 or k==n): return 1
%o A027219     elif (mod(n, 2)==0 and k==(n-2)/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
%o A027219     else: return T(n-1, k-1) + T(n-1, k)
%o A027219 [sum((k+1)*T(n,k) for k in (0..n)) for n in (0..30)] # _G. C. Greubel_, Jul 19 2019
%o A027219 (GAP)
%o A027219 T:= function(n, k)
%o A027219     if k=0 or k=n then return 1;
%o A027219     elif k=n-1 then return n;
%o A027219     elif (n mod 2)=0 and k=Int((n-2)/2) then return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k);
%o A027219     else return T(n-1, k-1) + T(n-1, k);
%o A027219     fi;
%o A027219   end;
%o A027219 List([0..20], n-> Sum([0..n], k-> (k+1)*T(n, k) )); # _G. C. Greubel_, Jul 19 2019
%Y A027219 Cf. A026736.
%K A027219 nonn
%O A027219 0,2
%A A027219 _Clark Kimberling_