OFFSET
0,2
COMMENTS
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
REFERENCES
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.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
N. J. Calkin, A curious binomial identity, Discr. Math., 131 (1994), 335-337.
Pietro Codara, Ottavio D'Antona, Francesco Marigo, Corrado Monti, Making simple proofs simpler, arXiv:1307.1348 [cs.MS], 2013.
Zachary Hamaker, Eric Marberg, Atoms for signed permutations, arXiv:1802.09805 [math.CO], 2018.
M. Hirschhorn, Calkin's binomial identity, Discr. Math., 159 (1996), 273-278.
Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, Discrete Math., 274 (2004), 331-342.
FORMULA
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).
From Robert Israel, Jan 13 2015: (Start)
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)
MAPLE
seq((n+2)*2^(2*n-1)-(n/2)*binomial(2*n, n), n=0..50); # Robert Israel, Jan 13 2015
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved