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A088009 Number of "sets of odd lists", cf. A000262. 9
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

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) )), recursively defined 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

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 April 17 09:05 EDT 2014. Contains 240634 sequences.