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A176334 Diagonal sums of number triangle A176331.

%I A176334 #11 Aug 08 2024 12:20:41
%S A176334 1,1,2,4,9,21,51,124,305,755,1879,4698,11792,29694,74984,189811,
%T A176334 481498,1223713,3115200,7942134,20275362,51823246,132604193,339644739,
%U A176334 870745187,2234208932,5737129623,14742751524,37909928908,97543380598
%N A176334 Diagonal sums of number triangle A176331.
%H A176334 G. C. Greubel, <a href="/A176334/b176334.txt">Table of n, a(n) for n = 0..1000</a>
%F A176334 a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(j,n-2k)*C(j,k)*(-1)^(n-k-j).
%F A176334 a(n) ~ phi^(2*n+3) / (4*5^(1/4)*sqrt(Pi*n)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Aug 08 2024
%p A176334 A176334 := proc(n)
%p A176334     add(add(binomial(j,n-2*k)*binomial(j,k)*(-1)^(n-k-j),j=0..n-k), k=0..floor(n/2)) ;
%p A176334 end proc: # _R. J. Mathar_, Feb 10 2015
%t A176334 T[n_, k_]:= T[n, k]= Sum[(-1)^(n-j)*Binomial[j, k]*Binomial[j, n-k], {j,0,n}]; Table[Sum[T[n-k, k], {k,0,Floor[n/2]}], {n,0,30}] (* _G. C. Greubel_, Dec 07 2019 *)
%o A176334 (PARI) T(n,k) = sum(j=0, n, (-1)^(n-j)*binomial(j, n-k)*binomial(j, k));
%o A176334 vector(30, n, sum(j=0, (n-1)\2, T(n-j-1,j)) ) \\ _G. C. Greubel_, Dec 07 2019
%o A176334 (Magma) T:= func< n,k | &+[(-1)^(n-j)*Binomial(j,n-k)*Binomial(j,k): j in [0..n]] >;
%o A176334 [(&+[T(n-k,k): k in [0..Floor(n/2)]]): n in [0..30]]; // _G. C. Greubel_, Dec 07 2019
%o A176334 (Sage)
%o A176334 @CachedFunction
%o A176334 def T(n, k): return sum( (-1)^(n-j)*binomial(j, n-k)*binomial(j, k) for j in (0..n))
%o A176334 [sum(T(n-k, k) for k in (0..floor(n/2))) for n in (0..30)] # _G. C. Greubel_, Dec 07 2019
%o A176334 (GAP)
%o A176334 T:= function(n,k)
%o A176334     return Sum([0..n], j-> (-1)^(n-j)*Binomial(j,k)*Binomial(j,n-k) );
%o A176334   end;
%o A176334 List([0..30], n-> Sum([0..Int(n/2)], j-> T(n-j,j) )); # _G. C. Greubel_, Dec 07 2019
%Y A176334 Cf. A176331, A176332, A176335.
%K A176334 easy,nonn
%O A176334 0,3
%A A176334 _Paul Barry_, Apr 15 2010