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 A003583 a(n) = (n+2)*2^(2*n-1) - (n/2)*binomial(2*n,n). 6
 1, 5, 26, 130, 628, 2954, 13612, 61716, 276200, 1223002, 5367676, 23383100, 101215576, 435712580, 1866667448, 7963424104, 33846062544, 143373104378, 605518549660, 2550438016812, 10716162617336 (list; graph; refs; listen; history; text; internal format)
 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 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. Sequence in context: A285905 A247491 A244617 * A033115 A033123 A331883 Adjacent sequences:  A003580 A003581 A003582 * A003584 A003585 A003586 KEYWORD nonn AUTHOR STATUS approved

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Last modified March 28 07:59 EDT 2020. Contains 333079 sequences. (Running on oeis4.)