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A362819 Number of ordered pairs of involutions on [n] that commute. 4
1, 1, 4, 10, 52, 196, 1216, 5944, 42400, 250912, 2008576, 13815616, 122074624, 950640640, 9158267392, 79258479616, 824644235776, 7823203807744, 87245790791680, 897748312609792, 10665239974537216, 118040852776093696, 1486172381689544704, 17572063073426206720, 233446797379437248512 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Two involutions x,y on [n] commute if x*y = y*x (i.e. x(y(i)) = y(x(i)) for i in [n]).
LINKS
FORMULA
a(n) = Sum_{k=0..floor(n/2)} A000085(n-2*k) * A000898(k) * binomial(n,2*k) * (2*k)! / (k!*2^k).
E.g.f.: exp(x + 3*x^2/2 + x^4/4).
PROG
(PARI) b(n, f) = {sum(k=0, n\2, f(k)*binomial(n, 2*k)*(2*k)!/(k!*2^k))}
a(n) = {b(n, k->b(n-2*k, j->1)*b(k, j->2^(k-j)))}
(PARI) seq(n)=Vec(serlaplace(exp(x + 3*x^2/2 + x^4/4 + O(x*x^n))))
CROSSREFS
Column k=2 of A362824.
A053529 is the corresponding sequence for all permutations.
Sequence in context: A208236 A032495 A109387 * A018844 A007027 A192444
KEYWORD
nonn
AUTHOR
Andrew Howroyd, May 05 2023
STATUS
approved

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Last modified July 25 00:10 EDT 2024. Contains 374585 sequences. (Running on oeis4.)