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%I #41 May 30 2018 13:54:57
%S 1,5,26,130,628,2954,13612,61716,276200,1223002,5367676,23383100,
%T 101215576,435712580,1866667448,7963424104,33846062544,143373104378,
%U 605518549660,2550438016812,10716162617336
%N a(n) = (n+2)*2^(2*n-1) - (n/2)*binomial(2*n,n).
%C a(n) gives the number of open partitions of a tree made of two chains with n points each, that share an added root. (An open partition pi of a tree T is a partition of the vertices of T with the property that, for each block B of pi, the upset of B is a union of blocks of pi.) - _Pietro Codara_, Jan 14 2015
%D Pietro Codara, Partitions of a finite partially ordered set, From Combinatorics to Philosophy: The Legacy of G.-C. Rota, Springer, New York (2009), 45-59.
%H G. C. Greubel, <a href="/A003583/b003583.txt">Table of n, a(n) for n = 0..1000</a>
%H N. J. Calkin, <a href="http://dx.doi.org/10.1016/0012-365X(94)90394-8">A curious binomial identity</a>, Discr. Math., 131 (1994), 335-337.
%H Pietro Codara, Ottavio D'Antona, Francesco Marigo, Corrado Monti, <a href="http://arxiv.org/abs/1307.1348">Making simple proofs simpler</a>, arXiv:1307.1348 [cs.MS], 2013.
%H Zachary Hamaker, Eric Marberg, <a href="https://arxiv.org/abs/1802.09805">Atoms for signed permutations</a>, arXiv:1802.09805 [math.CO], 2018.
%H M. Hirschhorn, <a href="http://dx.doi.org/10.1016/0012-365X(95)00086-C">Calkin's binomial identity</a>, Discr. Math., 159 (1996), 273-278.
%H Jun Wang and Zhizheng Zhang, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00206-1">On extensions of Calkin's binomial identities</a>, Discrete Math., 274 (2004), 331-342.
%F Main diagonal of correlation matrix of A055248. a(n) = Sum_{k=0..n} ( Sum_{m=k..n} binomial(n, m))^2 ). - _Paul Barry_, Jun 05 2003
%F Let S2 := (n, t)->add( k^t * (add( binomial(n, j), j=0..k))^2, k=0..n); a(n) = S2(n, 0).
%F From _Robert Israel_, Jan 13 2015: (Start)
%F G.f.: (1-2*x)/(1-4*x)^2 - x/(1 - 4*x)^(3/2).
%F E.g.f.: (2*x+1)*exp(4*x) - x*exp(2*x)*(I_0(2*x)+I_1(2*x)) where I_0 and I_1 are modified Bessel functions.
%F a(n) ~ 4^n*(n/2 - sqrt(n)/(2*sqrt(Pi)) + 1 + O(n^(-1/2))).
%F (End)
%p seq((n+2)*2^(2*n-1)-(n/2)*binomial(2*n,n), n=0..50); # _Robert Israel_, Jan 13 2015
%t Table[(n+2)*2^(2*n-1)-(n/2)*Binomial[2*n,n], {n,0,50}] (* _Pietro Codara_, Jan 14 2015 *)
%t Table[Sum[Sum[Binomial[n-1,k-1]Binomial[n-1,j-1]Min[k,j],{j,1,n}],{k,1 n}],{n,1,51}] (* _Pietro Codara_, Jan 14 2015 *)
%o (PARI) x='x+O('x^50); Vec((1-2*x)/(1-4*x)^2 - x/(1 - 4*x)^(3/2)) \\ _G. C. Greubel_, Feb 15 2017
%Y If the exponent E in a(n) = Sum_{m=0..n} (Sum_{k=0..m} C(n,k))^E is 1, 2, 3, 4, 5 we get A001792, A003583, A007403, A294435, A294436 respectively.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
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