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A088009
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Number of "sets of odd lists", cf. A000262.
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22
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1, 1, 1, 7, 25, 181, 1201, 10291, 97777, 1013545, 12202561, 151573951, 2173233481, 31758579997, 524057015665, 8838296029291, 164416415570401, 3145357419120721, 65057767274601217, 1391243470549894135, 31671795881695430521, 747996624368605997701
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OFFSET
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0,4
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COMMENTS
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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
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)
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LINKS
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FORMULA
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E.g.f.: exp(x/(1-x^2)).
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
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EXAMPLE
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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)
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MAPLE
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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):
# 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:
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MATHEMATICA
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a[n_] := SeriesCoefficient[ Exp[x/(1 - x^2) ], {x, 0, n}]*n!; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 24 2015 *)
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PROG
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(PARI)
x='x+O('x^33);
Vec(serlaplace(exp(x/(1-x^2))))
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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