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A184884 Diagonal sums of number triangle A184883.

%I A184884 #15 Sep 08 2022 08:45:55
%S A184884 1,1,2,6,11,27,60,132,301,669,1502,3370,7543,16919,37912,84968,190457,
%T A184884 426841,956698,2144238,4805827,10771315,24141588,54108332,121272549,
%U A184884 271806901,609198390,1365390546,3060236911,6858880431,15372743856,34454786384,77223188593,173079605553,387921692082,869445237846
%N A184884 Diagonal sums of number triangle A184883.
%H A184884 G. C. Greubel, <a href="/A184884/b184884.txt">Table of n, a(n) for n = 0..1000</a>
%H A184884 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,2,-1,1).
%F A184884 G.f.: (1-x^2)/(1-x-2*x^2-2*x^3+x^4-x^5).
%F A184884 a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..k} C(2*n-4*k,j)*C(k,j)*2^j.
%F A184884 a(n) = Sum_{k=0..floor(n/2)} Hypergeometric2F1([-k, 2*(k-n)], [1], 2). - _G. C. Greubel_, Nov 19 2021
%t A184884 LinearRecurrence[{1,2,2,-1,1}, {1,1,2,6,11}, 45] (* _G. C. Greubel_, Nov 19 2021 *)
%o A184884 (Magma)
%o A184884 A184883:= func< n,k | (&+[Binomial(k,j)*Binomial(2*(n-k), j)*2^j: j in [0..k]]) >;
%o A184884 A184884:= func< n | (&+[A184883(n, j): j in [0..Floor(n/2)]]) >;
%o A184884 [A184884(n): n in [0..40]]; // _G. C. Greubel_, Nov 19 2021
%o A184884 (Sage)
%o A184884 def A184883(n,k): return simplify( hypergeometric([-k, 2*(k-n)], [1], 2) )
%o A184884 def A184884(n): return sum( A184883(n, j) for j in (0..n//2) )
%o A184884 [A184884(n) for n in (0..40)] # _G. C. Greubel_, Nov 19 2021
%Y A184884 Cf. A183883.
%K A184884 nonn,easy
%O A184884 0,3
%A A184884 _Paul Barry_, Jan 24 2011