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%I #48 Jul 08 2022 12:17:01
%S 0,2,18,120,700,3780,19404,96096,463320,2187900,10161580,46558512,
%T 210924168,946454600,4212243000,18614102400,81746933040,357041751660,
%U 1551848136300,6715600122000,28947771052200,124337568995640,532337037821160,2272426880817600,9674281104930000
%N a(n) = (n + n^2)*binomial(2*n,n)/2.
%C For n > 0, also the number of one-sided prudent walks from (0,0) to (n,n), with n+2 east steps, 2 west steps and n north steps.
%H Bruno Berselli, <a href="/A119578/b119578.txt">Table of n, a(n) for n = 0..500</a>
%H Shanzhen Gao and Keh-Hsun Chen, <a href="http://worldcomp-proceedings.com/proc/p2014/FCS2696.pdf">Tackling Sequences From Prudent Self-Avoiding Walks</a>, FCS'14, The 2014 International Conference on Foundations of Computer Science.
%H S. Gao and H. Niederhausen, <a href="http://math.fau.edu/Niederhausen/HTML/Papers/Sequences%20Arising%20From%20Prudent%20Self-Avoiding%20Walks-February%2001-2010.pdf">Sequences Arising From Prudent Self-Avoiding Walks</a>, 2010.
%F a(n) = (n+1)*Gamma(2*n)/Gamma(n)^2 for n > 0. - _Shanzhen Gao_, Apr 26 2011
%F G.f.: 2 * x * (1 - x) / (1 - 4*x)^(5/2). - _Ilya Gutkovskiy_, Nov 17 2021
%F From _Amiram Eldar_, May 15 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 2*Pi^2/9 - 2*Pi/sqrt(3) + 2.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 4*sqrt(5)*log(phi) - 8*log(phi)^2 - 2, where phi is the golden ratio (A001622). (End)
%F D-finite with recurrence (-n+1)*a(n) +(5*n-1)*a(n-1) +2*(-2*n+3)*a(n-2)=0. - _R. J. Mathar_, Jul 08 2022
%F a(n) = A000217(n)*A000984(n). - _R. J. Mathar_, Jul 08 2022
%p [seq ((n+n^2)*(binomial(2*n,n))/2,n=0..29)];
%t Table[(n+n^2) Binomial[2n,n]/2,{n,0,30}] (* _Harvey P. Dale_, Jun 02 2016 *)
%o (Magma) [0] cat [ (n+1)*Factorial(2*n-1)/Factorial(n-1)^2: n in [1..23] ]; // _Klaus Brockhaus_, Apr 30 2011
%Y Cf. A000984, A001622, A085373.
%K nonn,easy
%O 0,2
%A _Zerinvary Lajos_, May 31 2006
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