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A084081 Sum of lists created by n substitutions k -> Range[k+1,0,-2] starting with {0}, counting down from k+1 to 0 step -2.

%I A084081 #9 Oct 18 2022 16:36:57
%S A084081 0,1,2,5,10,24,50,121,260,637,1400,3468,7752,19380,43890,110561,
%T A084081 253000,641355,1480050,3771885,8765250,22439040,52451256,134796060,
%U A084081 316663760,816540124,1926501200,4982228488,11798983280,30593078076,72690164850
%N A084081 Sum of lists created by n substitutions k -> Range[k+1,0,-2] starting with {0}, counting down from k+1 to 0 step -2.
%C A084081 Lengths of lists is A047749.
%H A084081 G. C. Greubel, <a href="/A084081/b084081.txt">Table of n, a(n) for n = 0..1000</a>
%F A084081 Equals A093951(n) - A047749(n).
%F A084081 From _G. C. Greubel_, Oct 17 2022: (Start)
%F A084081 a(2*n+1) = (3*n-1)*binomial[3*n+1, n]/((n+1)*(3*n+1)).
%F A084081 a(2*n) = 10*binomial(3*n+1, n-1)/(2*n+3). (End)
%e A084081 Lists {0}, {1}, {2, 0}, {3, 1, 1}, {4, 2, 0, 2, 0, 2, 0} sum to 0, 1, 2, 5, 10.
%t A084081 Plus@@@Flatten/@NestList[ # /. k_Integer :> Range[k+1, 0, -2]&, {0}, 8]
%t A084081 A084081[n_]:= If[EvenQ[n], 10*Binomial[(3*n+2)/2, (n-2)/2]/(n+3), 2*(3*n + 1)*Binomial[(3*n+5)/2, (n+1)/2]/((n+3)*(3*n+5))];
%t A084081 Table[A084081[n], {n, 40}] (* _G. C. Greubel_, Oct 17 2022 *)
%o A084081 (Magma)
%o A084081 F:=Floor; B:=Binomial;
%o A084081 function A084081(n)
%o A084081   if (n mod 2) eq 0 then return 10*B(F((3*n+2)/2), F((n-2)/2))/(n+3);
%o A084081   else return 2*(3*n+1)*B(F((3*n+5)/2), F((n+1)/2))/((n+3)*(3*n+5));
%o A084081   end if; return A084081;
%o A084081 end function;
%o A084081 [A084081(n): n in [0..40]]; // _G. C. Greubel_, Oct 17 2022
%o A084081 (SageMath)
%o A084081 def A084081(n):
%o A084081     if (n%2==0): return 10*binomial(int((3*n+2)/2), int((n-2)/2))/(n+3)
%o A084081     else: return 2*(3*n+1)*binomial(int((3*n+5)/2), int((n+1)/2))/((n+3)*(3*n+5))
%o A084081 [A084081(n) for n in range(40)] # _G. C. Greubel_, Oct 17 2022
%Y A084081 Cf. A047749, A093951.
%K A084081 nonn
%O A084081 0,3
%A A084081 _Wouter Meeussen_, May 11 2003