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A002739 a(n) = ((2*n-1)!/(2*n!*(n-2)!))*((n^3-3*n^2+2*n+2)/(n^2-1)).

%I A002739 M4732 N2024 #37 Oct 18 2022 19:15:18
%S A002739 1,10,91,651,4026,22737,120835,615043,3031678,14578928,68747966,
%T A002739 319075550,1461581460,6621579135,29718121635,132302508195,
%U A002739 584868588150,2569600678260,11227927978410,48822435838410,211370463290220,911509393468050,3916793943349326,16776146058210126,71641860657928876
%N A002739 a(n) = ((2*n-1)!/(2*n!*(n-2)!))*((n^3-3*n^2+2*n+2)/(n^2-1)).
%C A002739 The former name was "Coefficients for extrapolation".
%D A002739 J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 93.
%D A002739 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002739 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002739 G. C. Greubel, <a href="/A002739/b002739.txt">Table of n, a(n) for n = 2..1000</a>
%H A002739 J. Ser, <a href="/A002720/a002720_4.pdf">Les Calculs Formels des Séries de Factorielles</a>, Gauthier-Villars, Paris, 1933 [Local copy].
%H A002739 J. Ser, <a href="/A002720/a002720.pdf">Les Calculs Formels des Séries de Factorielles</a> (Annotated scans of some selected pages)
%F A002739 a(n) ~ 2^(2*n-5)*(8*n-33)*sqrt(n/Pi). - _Peter Luschny_, Jan 18 2020
%F A002739 From _G. C. Greubel_, Mar 23 2022: (Start)
%F A002739 a(n) = (1/4)*(n^3 - 3*n^2 + 2*n + 2)*A000108(n).
%F A002739 G.f.: (1 -9*x +21*x^2 +2*x^3)/(2*x*(1-4*x)^(5/2)) - (1 +x +x^2)/(2*x). (End)
%p A002739 t4 := n-> ((2*n-1)! /(2*n!*(n-2)!))*((n^3-3*n^2+2*n+2)/(n^2-1));
%p A002739 [seq(t4(n),n=2..40)];
%t A002739 Table[(n^3-3*n^2+2*n+2)*CatalanNumber[n]/4, {n,2,30}]
%o A002739 (Magma) [(n^3 -3*n^2 +2*n +2)*Catalan(n)/4: n in [2..30]]; // _G. C. Greubel_, Mar 23 2022
%o A002739 (Sage) [(n^3 -3*n^2 +2*n +2)*catalan_number(n)/4 for n in (2..30)] # _G. C. Greubel_, Mar 23 2022
%Y A002739 A diagonal of A331432.
%K A002739 nonn
%O A002739 2,2
%A A002739 _N. J. A. Sloane_
%E A002739 Entry revised by _N. J. A. Sloane_, Jan 18 2020