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A003583
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a(n) = (n+2)*2^(2*n-1) - (n/2)*binomial(2*n,n).
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12
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1, 5, 26, 130, 628, 2954, 13612, 61716, 276200, 1223002, 5367676, 23383100, 101215576, 435712580, 1866667448, 7963424104, 33846062544, 143373104378, 605518549660, 2550438016812, 10716162617336
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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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
Let S2 := (n, t)->add( k^t * (add( binomial(n, j), j=0..k))^2, k=0..n); a(n) = S2(n, 0).
G.f.: (1-2*x)/(1-4*x)^2 - x/(1 - 4*x)^(3/2).
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.
a(n) ~ 4^n*(n/2 - sqrt(n)/(2*sqrt(Pi)) + 1 + O(n^(-1/2))).
(End)
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MAPLE
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seq((n+2)*2^(2*n-1)-(n/2)*binomial(2*n, n), n=0..50); # Robert Israel, Jan 13 2015
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MATHEMATICA
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Table[(n+2)*2^(2*n-1)-(n/2)*Binomial[2*n, n], {n, 0, 50}] (* Pietro Codara, Jan 14 2015 *)
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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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