A003583 a(n) = (n+2)*2^(2*n-1) - (n/2)*binomial(2*n,n).

1, 5, 26, 130, 628, 2954, 13612, 61716, 276200, 1223002, 5367676, 23383100, 101215576, 435712580, 1866667448, 7963424104, 33846062544, 143373104378, 605518549660, 2550438016812, 10716162617336

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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.

Crossrefs

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.

Programs

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 *)

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

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)