A088026 -id:A088026 - cp's OEIS frontend

A088009 Number of "sets of odd lists", cf. A000262.

1, 1, 1, 7, 25, 181, 1201, 10291, 97777, 1013545, 12202561, 151573951, 2173233481, 31758579997, 524057015665, 8838296029291, 164416415570401, 3145357419120721, 65057767274601217, 1391243470549894135, 31671795881695430521, 747996624368605997701

Offset: 0

Vladeta Jovovic, Nov 02 2003

The Brauer algebra has a basis consisting of all graphs on the vertex set {1,...,2n} whose vertices all have degree 1. The multiplication is defined in Halverson and Ram. a(n) is also the number of idempotent basis elements (i.e., those satisfying b^2=b) of the Brauer algebra. - James East, Dec 27 2013
From Peter Bala, Nov 26 2017: (Start)
The sequence terms have the form 6*m + 1 (follows from the recurrence).
a(n+k) = a(n) (mod k) for all n and k. It follows that the sequence a(n) (mod k) is periodic with the exact period dividing k. For example, modulo 10 the sequence becomes 1, 1, 1, 7, 5, 1, 1, 1, 7, 5, ... with exact period 5. (End)

			From _R. J. Mathar_, Feb 01 2022 (Start):
Examples of partitions of elements {1,2,..n} into sets of lists where each list contains an odd number of elements:
n=1: One set where the element is the list.
n=2: One set where each of the 2 elements is its own list.
n=3: One set where each of the 3 elements is its own list, plus 6=3! sets of a list of all 3 elements.
n=4: One set where each of the 4 elements is its own list, plus 4*3! sets where one (4 choices) element is its own list and the remaining 3 elements are in another list.
n=5: One set where each of the 5 elements is its own list, plus 5!=120 sets where all 5 elements are in the same list, plus binomial(5,2)*3!=60 sets where two elements are in their own lists and the other 3 in a third list. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..446 (terms 0..200 from Alois P. Heinz)
I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, et al., Enumeration of idempotents in diagram semigroups and algebras, arXiv preprint arXiv:1408.2021 [math.GR], 2014.
T. Halverson and A. Ram, Partition algebras, arXiv:math/0401314 [math.RT], 2004.
T. Halverson and A. Ram, Partition algebras, European J. Combin. 26 (6) (2005) 869-921.

Cf. A052845, A088026, A000010, A000262, A168561, A318976.

T:= (n, k)-> `if`(n-k mod 2 = 0, binomial((n+k)/2, k), 0):
a:= n-> n! * add(T(n-1, k-1)/k!, k=0..n):
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 07 2011
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add((i->
      a(n-i)*binomial(n-1, i-1)*i!)(2*j+1), j=0..(n-1)/2))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Feb 01 2022

a[n_] := SeriesCoefficient[ Exp[x/(1 - x^2) ], {x, 0, n}]*n!; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 24 2015 *)

x='x+O('x^33);
Vec(serlaplace(exp(x/(1-x^2))))
/* Joerg Arndt, Mar 09 2011 */

E.g.f.: exp(x/(1-x^2)).
a(n) = n!*Sum_{k=1..n} A168561(n-1,k-1)/k!. - Vladimir Kruchinin, Mar 07 2011
E.g.f.: 1 + x/(G(0)-x) where G(k)= (1-x^2)*k + 1+x-x^2 - x*(1-x^2)*(k+1)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Aug 02 2012
E.g.f.: 1 + x/(1+x)*(G(0) - 1) where G(k) = 1 + 1/(1+x^2)/(k+1)/(1-x/(x+(1)/G(k+1))), (continued fraction). - Sergei N. Gladkovskii, Feb 04 2013
a(n) ~ 2^(-3/4)*n^(n-1/4)*exp(sqrt(2*n)-n) * (1-11/(24*sqrt(2*n))). - Vaclav Kotesovec, Aug 10 2013
D-finite with recurrence a(n) = a(n-1) + 2*(n-2)*(n-1)*a(n-2) + (n-2)*(n-1)*a(n-3) - (n-4)*(n-3)*(n-2)*(n-1)*a(n-4). - Vaclav Kotesovec, Aug 10 2013
E.g.f.: Product_{n >= 1} (1 + x^n)^(phi(n)/n) = Product_{n >= 0} ( (1 + x^(2*n+1))/(1 - x^(2*n+1)) )^( phi(2*n+1)/(4*n + 2) ), where phi(n) = A000010(n) is the Euler totient function. Cf. A066668 and A000262. - Peter Bala, Jan 01 2014
E.g.f.: Product_{k>0} exp(x^(2*k-1)). - Seiichi Manyama, Oct 10 2017

Prepended a(0)=1 by Joerg Arndt, Jul 29 2012

A206703 Triangular array read by rows. T(n,k) is the number of partial permutations (injective partial functions) of {1,2,...,n} that have exactly k elements in a cycle. The k elements are not necessarily in the same cycle. A fixed point is considered to be in a cycle.

Original entry on oeis.org

1, 1, 1, 3, 2, 2, 13, 9, 6, 6, 73, 52, 36, 24, 24, 501, 365, 260, 180, 120, 120, 4051, 3006, 2190, 1560, 1080, 720, 720, 37633, 28357, 21042, 15330, 10920, 7560, 5040, 5040, 394353, 301064, 226856, 168336, 122640, 87360, 60480, 40320, 40320

Offset: 0

Geoffrey Critzer, Feb 11 2012

			     1;
     1,     1;
     3,     2,     2;
    13,     9,     6,     6;
    73,    52,    36,    24,    24;
   501,   365,   260,   180,   120,  120;
  4051,  3006,  2190,  1560,  1080,  720,   720;
  ...

Mohammad K. Azarian, On the Fixed Points of a Function and the Fixed Points of its Composite Functions, International Journal of Pure and Applied Mathematics, Vol. 46, No. 1, 2008, pp. 37-44. Mathematical Reviews, MR2433713 (2009c:65129), March 2009. Zentralblatt MATH, Zbl 1160.65015.
Mohammad K. Azarian, Fixed Points of a Quadratic Polynomial, Problem 841, College Mathematics Journal, Vol. 38, No. 1, January 2007, p. 60. Solution published in Vol. 39, No. 1, January 2008, pp. 66-67.

Alois P. Heinz, Rows n = 0..140, flattened
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 132.

Columns k = 0..1 give: A000262, A006152.
Main diagonal gives A000142.
Row sums give A002720.
T(2n,n) gives A088026.
Cf. A002720, A052852, A103194, A105219, A331725.

b:= proc(n) option remember; `if`(n=0, 1, add((p-> p+x^j*
      coeff(p, x, 0))(b(n-j)*binomial(n-1, j-1)*j!), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Feb 19 2022

nn = 7; a = 1/(1 - x); ay = 1/(1 - y x); f[list_] := Select[list, # > 0 &]; Map[f, Range[0, nn]! CoefficientList[Series[Exp[a x] ay, {x, 0, nn}], {x, y}]] // Flatten

E.g.f.: exp(x/(1-x))/(1-y*x).
From Alois P. Heinz, Feb 19 2022: (Start)
Sum_{k=1..n} T(n,k) = A052852.
Sum_{k=0..n} k * T(n,k) = A103194(n).
Sum_{k=0..n} (n-k) * T(n,k) =  A105219(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A331725(n). (End)

A065140 a(n) = 2^n(2n)!.

Original entry on oeis.org

1, 4, 96, 5760, 645120, 116121600, 30656102400, 11158821273600, 5356234211328000, 3278015337332736000, 2491291656372879360000, 2301953490488540528640000, 2541356653499348743618560000, 3303763649549153366704128000000, 4995290638118319890456641536000000

Offset: 0

Karol A. Penson, Oct 16 2001

From Enrique Navarrete, Aug 29 2025: (Start)
For n > 0, 1/2*a(n) is the number of ways to seat 2*n people on linearly ordered benches placing an even number of people (>=2) on each bench.
For example, 1/2*a(4)=322560 since the number of ways are (number of people in parentheses):
  1 bench (8): 40320 ways;
  2 benches (6,2): 80640 ways;
  2 benches (4,4): 40320 ways;
  3 benches (4,2,2): 120960 ways;
  4 benches (2,2,2,2): 40320 ways.
If the benches are not linearly ordered the number of ways is A088026.
If we seat an odd number of people on linearly ordered benches the number of ways is A005443. (End)

Harry J. Smith, Table of n, a(n) for n = 0..100

Cf. A000680, A088026.

Table[2^n (2n)!,{n,0,15}] (* Harvey P. Dale, Nov 28 2011 *)

{ for (n=0, 100, write("b065140.txt", n, " ", 2^n*(2*n)!) ) } \\ Harry J. Smith, Oct 11 2009

Hypergeometric generating function, in Maple notation: 1/sqrt(1-8*x), i.e., a(0)=1, a(n)=round(evalf(subs(x=0, n!*diff(1/(sqrt(1-8*x)), x$n)))), for n>=1.
Integral representation as n-th moment of a positive function on a positive half-axis: a(n) = Integral_{x>=0} x^n*exp(-sqrt(x/2))/(2*sqrt(2*x)) dx, for n>=0.
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 4*x*(k+1)*(2*k+1)/(4*x*(k+1)*(2*k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 07 2013
From Amiram Eldar, Aug 05 2020: (Start)
Sum_{n>=0} 1/a(n) = cosh(sqrt(2)/2).
Sum_{n>=0} (-1)^n/a(n) = cos(sqrt(2)/2). (End)
From Alexandre Herrera, Apr 18 2025: (Start)
Sum_{n>=0} x^(4*n)*(-1)^(n)/a(2n) = cos(x/2)*cosh(x/2).
Sum_{n>=0} x^(4*n+2)*(-1)^(n)/a(2n+1) = sin(x/2)*sinh(x/2).
Sum_{n>=0} x^(2*n)*(-1)^(n)/a(n) = cos(x*sqrt(2)/2).
Sum_{n>=0} x^(2*n)/a(n) = cosh(x*sqrt(2)/2). (End)

A096965 Number of sets of even number of even lists, cf. A000262.

Original entry on oeis.org

1, 1, 1, 7, 37, 241, 2101, 18271, 201097, 2270017, 29668681, 410815351, 6238931821, 101560835377, 1765092183037, 32838929702671, 644215775792401, 13441862819232001, 293976795292186897, 6788407001443004647, 163735077313046119861, 4142654439686285737201

Offset: 0

Vladeta Jovovic, Aug 18 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000262, A088026, A088009, A096939, A349776.

a:= proc(n) option remember;  `if`(n<4, [1$3, 7][n+1], ((2*n-3)
      *a(n-1)+(n-1)*(2*n^2-8*n+7)*a(n-2) + (n-2)*(n-1)*(2*n-5)
      *a(n-3)-(n-4)*(n-3)*(n-2)^2*(n-1)*a(n-4))/(n-2))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 01 2021

Drop[ Range[0, 20]! CoefficientList[ Series[ Exp[(x/(1 - x^2))]Cosh[x^2/(1 - x^2)], {x, 0, 20}], x], 1] (* Robert G. Wilson v, Aug 19 2004 *)

E.g.f.: exp(x/(1-x^2))*cosh(x^2/(1-x^2)).
a(n) = (n!*sum(m=floor((n+1)/2)..n, (binomial(n-1,2*m-n-1))/(2*m-n)!)). - Vladimir Kruchinin, Mar 10 2013
Recurrence: (n-2)*a(n) = (2*n-3)*a(n-1) + (n-1)*(2*n^2 - 8*n + 7)*a(n-2) + (n-2)*(n-1)*(2*n-5)*a(n-3) - (n-4)*(n-3)*(n-2)^2*(n-1)*a(n-4). - Vaclav Kotesovec, Sep 29 2013
a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n-1/4)/(2*sqrt(2)) * (1-5/(48*sqrt(n))). - Vaclav Kotesovec, Sep 29 2013
From Alois P. Heinz, Dec 01 2021: (Start)
a(n) = A000262(n) - A096939(n).
a(n) = |Sum_{k=0..n} (-1)^k * A349776(n,k)|. (End)

More terms from Robert G. Wilson v, Aug 19 2004

a(0)=1 prepended by Alois P. Heinz, Dec 01 2021

A096939 Number of sets of odd number of even lists, cf. A000262.

Original entry on oeis.org

0, 2, 6, 36, 260, 1950, 19362, 193256, 2326536, 29272410, 413257790, 6231230412, 101415565836, 1769925341366, 32734873484250, 646218442877520, 13404753632014352, 294656673023216946, 6775966692145553526

Offset: 1

Vladeta Jovovic, Aug 18 2004

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A000262, A088026, A088009, A096965.

Drop[ Range[0, 20]! CoefficientList[ Series[ Exp[(x/(1 - x^2))] Sinh[x^2/(1 - x^2)], {x, 0, 20}], x], 1] (* Robert G. Wilson v, Aug 19 2004 *)

E.g.f.: exp(x/(1-x^2))*sinh(x^2/(1-x^2)).
Recurrence: (n-2)*a(n) = (2*n-3)*a(n-1) + (n-1)*(2*n^2 - 8*n + 7)*a(n-2) + (n-2)*(n-1)*(2*n-5)*a(n-3) - (n-4)*(n-3)*(n-2)^2*(n-1)*a(n-4). - Vaclav Kotesovec, Sep 29 2013
a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n-1/4)/(2*sqrt(2)) * (1-5/(48*sqrt(n))). - Vaclav Kotesovec, Sep 29 2013
a(n) = A000262(n) - A096965(n). - Alois P. Heinz, Dec 01 2021

More terms from Robert G. Wilson v, Aug 19 2004

A102760 Number of partitions of n-set into "lists", in which every even list appears an even number of times, cf. A000262.

Original entry on oeis.org

1, 1, 1, 7, 37, 241, 1381, 13231, 140617, 1483777, 16211881, 217551511, 3384215341, 50221272817, 782154787597, 13913712591871, 272739557719441, 5282625708305281, 106588332600443857, 2354480141600267047, 56238135934525073461, 1338131691952924913521

Offset: 0

Vladeta Jovovic, Feb 10 2005

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..445

Cf. A003483, A006950, A052565, A088026.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(i::even and j::odd, 0, b(n-i*j, i-1)*
      multinomial(n, n-i*j, i$j)/j!*i!^j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[EvenQ[i] && OddQ[j], 0, b[n-i*j, i- 1] * multinomial[n, Join[{n - i*j}, Array[i &, j]]]/j!*i!^j], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)

E.g.f.: exp(x/(1-x^2))*Product_{k>0} cosh(x^(2*k)).

a(0)=1 prepended by Alois P. Heinz, May 10 2016

cp's OEIS Frontend

A088009 Number of "sets of odd lists", cf. A000262.

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A206703 Triangular array read by rows. T(n,k) is the number of partial permutations (injective partial functions) of {1,2,...,n} that have exactly k elements in a cycle. The k elements are not necessarily in the same cycle. A fixed point is considered to be in a cycle.

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A065140 a(n) = 2^n*(2*n)!.

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A096965 Number of sets of even number of even lists, cf. A000262.

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Maple

Mathematica

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Extensions

A096939 Number of sets of odd number of even lists, cf. A000262.

Views

Author

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Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A102760 Number of partitions of n-set into "lists", in which every even list appears an even number of times, cf. A000262.

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Author

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Maple

Mathematica

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Extensions

A065140 a(n) = 2^n(2n)!.