A229243 -id:A229243 - cp's OEIS frontend

A066223 Bisection of A000085.

1, 2, 10, 76, 764, 9496, 140152, 2390480, 46206736, 997313824, 23758664096, 618884638912, 17492190577600, 532985208200576, 17411277367391104, 606917269909048576, 22481059424730751232, 881687990282453393920, 36494410645223834692096, 1589659519990672490875904

Offset: 0

N. J. A. Sloane, Dec 19 2001

Number of tableaux on 2n elements. - Roberto E. Martinez II, Jan 09 2002
a(n) = number of ways to connect 2n points labeled 1,2,...,2n in a line with 0 or more arcs such that at most one arc leaves each point. For example, with arcs separated by dashes, a(2)=10 counts {} (no arcs), 12, 13, 14, 23, 24, 34, 12-34, 13-24, 14-23. - David Callan, Sep 18 2007
a(n) = A229223(2n,2) = A229243(2,n). - Alois P. Heinz, Sep 17 2013

S. Chowla, The asymptotic behavior of solutions of difference equations, in Proceedings of the International Congress of Mathematicians (Cambridge, MA, 1950), Vol. I, 377, Amer. Math. Soc., Providence, RI, 1952.

Alois P. Heinz, Table of n, a(n) for n = 0..200
T. Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
I. Dolinka, J. East and R. D. Gray, Motzkin monoids and partial Brauer monoids, arXiv preprint arXiv:1512.02279 [math.GR], 2015.
Michael Torpey, Semigroup congruences: computational techniques and theoretical applications, Ph.D. Thesis, University of St. Andrews (Scotland, 2019).

Cf. A066224.

a:= proc(n) option remember; `if`(n<2, n+1,
      (4*n-2)*a(n-1)-2*(n-1)*(2*n-3)*a(n-2))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 17 2013

NumberOfTableaux[2n]
a[n_] := a[n] = If[n<2, n+1, (4*n-2)*a[n-1] - 2*(n-1)*(2*n-3)*a[n-2]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 13 2014, after Alois P. Heinz *)
Table[(-2)^n HypergeometricU[-n, 1/2, -(1/2)], {n, 0, 90}] (* Emanuele Munarini, Aug 31 2017 *)

a(n)=sum(k=0,n,binomial(2*n,2*k)*prod(i=1,k,2*i-1))

a(n)=if(n<0, 0, n*=2; n!*polcoeff(exp(x+x^2/2+x*O(x^n)),n))

a(n) = sum(k=0, n, C(2n, 2*k)*(2k-1)!!). - Benoit Cloitre, May 01 2003
a(n) = n!*2^n*LaguerreL(n, -1/2, -1/2). - Vladeta Jovovic, May 10 2003
E.g.f.: cosh(x)*exp(x^2/2) (with interpolated zeros) - Paul Barry, May 26 2003
E.g.f.: exp(x/(1-2*x))/sqrt(1-2*x). - Paul Barry, Apr 12 2010
a(n) = (1/sqrt(2*pi))*Int((1+x)^(2*n)*exp(-x^2/2),x,-infinity,infinity). - Paul Barry, Apr 21 2010
Conjecture: a(n) +2*(-2*n+1)*a(n-1) +2*(n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 24 2012
Remark: the above conjectured recurrence is true and can be obtained by the e.g.f. - Emanuele Munarini, Aug 31 2017
a(n) ~ n^n*2^(n-1/2)*exp(-n+sqrt(2*n)-1/4) * (1 + 7/(24*sqrt(2*n))). - Vaclav Kotesovec, Jun 22 2013

More terms from Roberto E. Martinez II, Jan 09 2002

A229228 Number of set partitions of {1,...,2n} into sets of size at most n.

Original entry on oeis.org

1, 1, 10, 166, 3795, 112124, 4163743, 190168577, 10468226150, 681863474058, 51720008131148, 4506628734688128, 445956917001833090, 49631199898024188422, 6160538225093750695800, 846748983034696433927334, 128064669166890886264698699, 21195039362681903376709497444

Offset: 0

Alois P. Heinz, Sep 16 2013

nonn

			a(2) = 10: 1/2/3/4, 12/3/4, 13/2/4, 14/2/3, 1/23/4, 1/24/3, 1/2/34, 12/34, 13/24, 14/23.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k=2 of A229243.

Cf. A229223.

G:= proc(n, k) option remember; local j; if k>n then G(n, n)
      elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
      for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j, k) od; % fi
    end:
a:= n-> G(2*n, n):
seq(a(n), n=0..20);

G[n_, k_] := G[n, k] = If[n == 0, 1, If[k < 1, 0, Sum[G[n - k*j, k - 1]*n!/ k!^j/(n - k*j)!/j!, {j, 0, n/k}]]];
Table[G[2n, n], {n, 0, 20}] (* Jean-François Alcover, May 21 2018, translated from Maple *)

a(n) = (2n)! * [x^(2n)] exp(Sum_{j=1..n} x^j/j!).

a(n) = A229223(2n,n).

A229229 Number of set partitions of {1,...,n^2} into sets of size at most n.

Original entry on oeis.org

1, 1, 10, 12644, 6631556521, 3282701194678476257, 3025262978042089315465899013351, 9292286146024114784457467780130028866860171013, 158655194198118596873150397161518177395553186289541468458000908304

Offset: 0

Alois P. Heinz, Sep 16 2013

nonn

			a(2) = 10: 1/2/3/4, 12/3/4, 13/2/4, 14/2/3, 1/23/4, 1/24/3, 1/2/34, 12/34, 13/24, 14/23.

Alois P. Heinz, Table of n, a(n) for n = 0..24

Main diagonal of A229243.

Cf. A229223.

G:= proc(n, k) option remember; local j; if k>n then G(n, n)
      elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
      for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j, k) od; % fi
    end:
a:= n-> G(n^2, n):
seq(a(n), n=0..10);

G[n_, k_] := G[n, k] = Module[{j, pc}, Which[k>n, G[n, n], n==0, 1, k<1, 0, True, pc = G[n-k, k]; For[j = k-1, j >= 1, j--, pc = pc*(n-j)/j + G[n-j, k]]; pc]]; a[n_] := G[n^2, n]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Feb 15 2017, translated from Maple *)

a(n) = (n^2)! * [x^(n^2)] exp(Sum_{j=1..n} x^j/j!).

a(n) = A229223(n^2,n).

A229413 Number of set partitions of {1,...,3n} into sets of size at most n.

Original entry on oeis.org

1, 1, 76, 12644, 3305017, 1245131903, 654277037674, 467728049807348, 443694809361207824, 544852927413901502514, 846359710104516310431744, 1629392161877794034658847500, 3819592516111353522143561652540, 10738740219595085951726635839975852

Offset: 0

Alois P. Heinz, Sep 22 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..150

Column k=3 of A229243.

Cf. A229223.

G:= proc(n, k) option remember; local j; if k>n then G(n, n)
      elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
      for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j, k) od; % fi
    end:
a:= n-> G(3*n, n):
seq(a(n), n=0..20);

G[n_, k_] := G[n, k] = Module[{j, g}, Which[k > n, G[n, n], n == 0, 1, k < 1, 0, True, g = G[n - k, k]; For[j = k - 1, j >= 1, j--, g = g(n-j)/j + G[n - j, k]]; g]];
a[n_] := G[3n, n];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)

a(n) = (3n)! * [x^(3n)] exp(Sum_{j=1..n} x^j/j!).

a(n) = A229223(3n,n).

A229414 Number of set partitions of {1,...,3n} into sets of size at most 3.

Original entry on oeis.org

1, 5, 166, 12644, 1680592, 341185496, 97620050080, 37286121988256, 18280749571449664, 11168256342434121152, 8306264068494786829696, 7380771881944947770497280, 7715405978050522488223499776, 9365880670184268387214967727104, 13058232187415887547449498864463872

Offset: 0

Alois P. Heinz, Sep 22 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Row n=3 of A229243.

a:= proc(n) option remember; `if`(n<3, [1, 5, 166][n+1],
      ((108*n^2-72*n+4)*a(n-1)-6*(n-1)*(3*n-5)*(27*n^2-48*n+10)*a(n-2)
       +9*(n-1)*(n-2)*(3*n-1)*(3*n-7)*(3*n-5)*(3*n-8)*a(n-3))/8)
    end:
seq(a(n), n=0..20);

G[n_, k_] := G[n, k] = Module[{j, g}, Which[k > n, G[n, n], n == 0, 1, k < 1, 0, True, g = G[n - k, k]; For[j = k - 1, j >= 1, j--, g = g(n-j)/j + G[n - j, k]]; g]];
a[n_] := G[3n, 3];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz in A229243 *)

a(n) = (3n)! * [x^(3n)] exp(x + x^2/2 + x^3/6).

a(n) = A001680(3n) = A229223(3n,3).

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A066223 Bisection of A000085.

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A229228 Number of set partitions of {1,...,2n} into sets of size at most n.

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A229229 Number of set partitions of {1,...,n^2} into sets of size at most n.

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A229413 Number of set partitions of {1,...,3n} into sets of size at most n.

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A229414 Number of set partitions of {1,...,3n} into sets of size at most 3.

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Maple

Mathematica

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