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A088009 Number of "sets of odd lists", cf. A000262. 10
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 (ie, those satisfying b^2=b) of the Brauer algebra. - James East, Dec 27 2013

LINKS

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

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, 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

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.

Sequence in context: A151491 A208425 A191237 * A208823 A197913 A215058

Adjacent sequences:  A088006 A088007 A088008 * A088010 A088011 A088012

KEYWORD

nonn

AUTHOR

Vladeta Jovovic, Nov 02 2003

EXTENSIONS

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

STATUS

approved

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Last modified December 4 21:09 EST 2016. Contains 278755 sequences.