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A132745 Row sums of (A008550 formatted as a triangular array).

%I A132745 #8 Feb 16 2021 17:48:33
%S A132745 1,2,3,5,11,32,114,467,2130,10642,57629,335381,2082582,13716502,
%T A132745 95352529,696790819,5334094259,42649956716,355261078352,3075741331481,
%U A132745 27620835538407,256810928552476,2468108094076860,24481671811988907,250296546308500181,2634309876797453868,28509045368598994348
%N A132745 Row sums of (A008550 formatted as a triangular array).
%H A132745 G. C. Greubel, <a href="/A132745/b132745.txt">Table of n, a(n) for n = 0..500</a>
%F A132745 From _G. C. Greubel_, Feb 16 2021: (Start)
%F A132745 a(n) = Sum_{k=0..n} Hypergeometric2F1([1-n+k, k-n], [2], k).
%F A132745 a(n) = Sum_{k=0..n} Hypergeometric2F1([1-k, -k], [2], n-k).
%F A132745 a(n) = 1 + Sum_{k=1..n} Sum_{j=0..k-1} binomial(k,j)^2 * ((k-j)*(n-k)^j/(k*(j+1))).
%F A132745 a(n) = 1 + Sum_{k=1..n} Sum_{j=0..k-1} A001263(k, k-j) * (n-k)^j. (End)
%t A132745 Table[Sum[Hypergeometric2F1[1-k, -k, 2, n-k], {k,0,n}], {n,0,30}] (* _G. C. Greubel_, Feb 16 2021 *)
%o A132745 (Sage)
%o A132745 def A243631(n,k): return 1 if n==0 else sum( binomial(n,j)^2*k^j*(n-j)/(n*(j+1)) for j in [0..n-1])
%o A132745 def A132745(n): return sum( A243631(k, n-k) for k in [0..n] )
%o A132745 [A132745(n) for n in [0..30]] # _G. C. Greubel_, Feb 16 2021
%o A132745 (Magma)
%o A132745 A243631:= func< n,k | n eq 0 select 1 else (&+[ Binomial(n,j)^2*k^j*(n-j)/(n*(j+1)): j in [0..n-1]]) >;
%o A132745 A132745:= func< n | (&+[A243631(k,n-k): k in [0..n]]) >;
%o A132745 [A132745(n): n in [0..30]]; // _G. C. Greubel_, Feb 16 2021
%Y A132745 Cf. A001263, A008550, A243631.
%K A132745 nonn
%O A132745 0,2
%A A132745 _Philippe Deléham_, Nov 21 2007
%E A132745 Terms a(11) onward added by _G. C. Greubel_, Feb 16 2021