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
Examples
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)
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 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.
Programs
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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 # 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
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Mathematica
a[n_] := SeriesCoefficient[ Exp[x/(1 - x^2) ], {x, 0, n}]*n!; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 24 2015 *) -
PARI
x='x+O('x^33); Vec(serlaplace(exp(x/(1-x^2)))) /* Joerg Arndt, Mar 09 2011 */
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
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
Extensions
Prepended a(0)=1 by Joerg Arndt, Jul 29 2012
Comments