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%I #50 Sep 08 2022 08:46:09
%S -1,1,7,25,87,329,1359,6001,27759,132689,649815,3242377,16421831,
%T 84196761,436129183,2278835681,11996748255,63568974241,338777252263,
%U 1814623238137,9763917858359,52750451120361,286036294786287,1556185889290065,8492182185653327,46471113779766769
%N a(n) = Sum_{k=0..n} C(n, k)*C(n+k, k)/(2k-1), where C(n, k) denotes the binomial coefficient n!/(k!*(n-k)!).
%C Note that a(n) is always an integer since C(n,k)*C(n+k,k) = C(n+k,2k)*C(2k,k), and C(2k,k)/(2k-1) = 2*Catalan(k-1) for k > 0.
%C Conjecture: The sequence a(n+1)/a(n) (n = 3,4,...) is strictly increasing to the limit 3+2*sqrt(2), and the sequence a(n+1)^(1/(n+1))/a(n)^(1/n) (n = 5,6,...) is strictly decreasing to the limit 1.
%H Zhi-Wei Sun, <a href="/A245769/b245769.txt">Table of n, a(n) for n = 0..200</a>
%H Victor J. W. Guo, and Ji-Cai Liu, <a href="https://doi.org/10.1016/j.jnt.2015.04.024">Proof of a conjecture of Z.-W. Sun on the divisibility of a triple sum</a>, Journal of Number Theory, Volume 156, November 2015, Pages 154-160. Rn is a(n).
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1408.5381">A new kind of numbers and their arithmetic properties</a>, arXiv:1408.5381 [math.NT], 2017.
%F Recurrence (obtained via the Zeilberger algorithm): (n+1)*a(n) - (7*n+15)*a(n+1) + (7*n+13)*a(n+2) - (n+3)*a(n+3) = 0.
%F a(n) ~ A006318(n)/2 as n tends to the infinity, thus a(n)^(1/n) has the limit 3+2*sqrt(2).
%F 0 = +a(n)*(+a(n+1) -15*a(n+2) +13*a(n+3) -3*a(n+4)) +a(n+1)*(+a(n+1) +50*a(n+2) -34*a(n+3) +13*a(n+4)) +a(n+2)*(-63*a(n+2) +50*a(n+3) -15*a(n+4)) +a(n+3)*(+a(n+3) +a(n+4)) for all n in Z. - _Michael Somos_, Aug 24 2014
%e a(2) = 7 since sum_{k=0,1,2}C(2,k)*C(2+k,k)/(2k-1) = -1 + 6 + 6/3 = 7.
%t a[n_]:=Sum[Binomial[n,k]Binomial[n+k,k]/(2k-1),{k,0,n}]; Table[a[n],{n, 0, 25}]
%o (PARI) for(n=0,25, print1(sum(k=0,n, binomial(n,k)*binomial(n+k,k)/(2*k -1)), ", ")) \\ _G. C. Greubel_, Aug 05 2018
%o (Magma) [(&+[Binomial(n,k)*Binomial(n+k,k)/(2*k-1): k in [0..n]]): n in [0..25]]; // _G. C. Greubel_, Aug 05 2018
%Y Cf. A000108, A006318.
%K sign
%O 0,3
%A _Zhi-Wei Sun_, Aug 24 2014
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