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A088009 Number of "sets of odd lists", cf. A000262. 15
1, 1, 1, 7, 25, 181, 1201, 10291, 97777, 1013545, 12202561, 151573951, 2173233481, 31758579997, 524057015665, 8838296029291, 164416415570401, 3145357419120721, 65057767274601217, 1391243470549894135, 31671795881695430521, 747996624368605997701 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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)

LINKS

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, A. Ram, Partition algebras, arXiv:math/0401314 [math.RT], 2004.

T. Halverson, A. Ram, Partition algebras, European J. Combin. 26 (6) (2005) 869-921.

FORMULA

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

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

MAPLE

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

MATHEMATICA

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

PROG

(PARI)

x='x+O('x^33);

Vec(serlaplace(exp(x/(1-x^2))))

/* Joerg Arndt, Mar 09 2011 */

CROSSREFS

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

Sequence in context: A208425 A191237 * A293532 A208823 A197913 A215058

Adjacent sequences:  A088006 A088007 A088008 * A088010 A088011 A088012

KEYWORD

nonn,easy,changed

AUTHOR

Vladeta Jovovic, Nov 02 2003

EXTENSIONS

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

STATUS

approved

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Last modified December 17 16:25 EST 2017. Contains 296119 sequences.