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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 Andrew Howroyd, Table of n, a(n) for n = 0..200 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. Cf. A000085, A000898, A181162, A362820, A362825. Sequence in context: A208236 A032495 A109387 * A018844 A007027 A192444 Adjacent sequences: A362816 A362817 A362818 * A362820 A362821 A362822 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.)