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A124642 Antidiagonal sums of A096465.

%I A124642 #20 May 01 2021 07:38:15
%S A124642 1,1,2,3,5,9,15,29,50,99,176,351,638,1275,2354,4707,8789,17577,33099,
%T A124642 66197,125477,250953,478193,956385,1830271,3660541,7030571,14061141,
%U A124642 27088871,54177741,104647631,209295261,405187826,810375651,1571990936,3143981871,6109558586,12219117171,23782190486,47564380971,92705454896
%N A124642 Antidiagonal sums of A096465.
%C A124642 Apparently bisections give A024718 and A006134 and are related to A078478, A100066 and A105848.
%H A124642 G. C. Greubel, <a href="/A124642/b124642.txt">Table of n, a(n) for n = 0..1000</a>
%F A124642 Conjecture: G.f.: -(1/2)*z*(2*z+(1-4*z^2)^(1/2)+1)/(1-4*z^2)^(1/2)/(z^2-1). -  Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009
%F A124642 From _G. C. Greubel_, Apr 30 2021: (Start)
%F A124642 a(n) = (1 + (-1)^n)/2 + Sum_{j=0..floor((n-1)/2)} Sum_{k=0..j} (n-2*j)*binomial(n -2*k, n-k-j)/(n-2*k).
%F A124642 a(n) = Sum_{j=0..floor(n/2)} Sum_{k=0..j} ((n-2*j)/(n-k-j))*binomial(n-2*k, n-k-j). (End)
%t A124642 a[_, 0]=1; a[n_, n_]=1; a[n_, m_]:= a[n, m] = a[n-1, m] + a[n, m-1]; a[n_, m_] /; n<0 || m>n = 0; Table[ Sum[a[n-m, m], {m,0,n}], {n,0,45}] (* _Jean-François Alcover_, Dec 17 2012 *)
%t A124642 a[n_]:= a[n]= (1+(-1)^n)/2 + Sum[(n-2*j)*Binomial[n-2*k, n-k-j]/(n-2*k), {j,0,(n-1)/2}, {k,0,j}]; Table[a[n], {n,0,45}] (* _G. C. Greubel_, Apr 30 2021 *)
%o A124642 (Magma)
%o A124642 a:= func< n | n eq 0 select 1 else (1+(-1)^n)/2 + (&+[ (&+[ ((n-2*j)/(n-2*k))*Binomial(n-2*k, n-k-j) : k in [0..j]]) : j in [0..Floor((n-1)/2)]]) >;
%o A124642 [a(n): n in [0..45]]; // _G. C. Greubel_, Apr 30 2021
%o A124642 (Sage)
%o A124642 def a(n): return (1+(-1)^n)/2 + sum( sum( ((n-2*j)/(n-2*k))*binomial(n-2*k, n-k-j) for k in (0..j)) for j in (0..(n-1)//2))
%o A124642 [a(n) for n in (0..45)] # _G. C. Greubel_, Apr 30 2021
%Y A124642 Cf. A006134, A024718, A078478, A100066, A105848.
%K A124642 nonn
%O A124642 0,3
%A A124642 _Gerald McGarvey_, Dec 21 2006
%E A124642 Offset changed by _Reinhard Zumkeller_, Jul 12 2012
%E A124642 Terms a(18) onward added by _G. C. Greubel_, Apr 30 2021