This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A373339 #22 Jun 28 2024 22:51:45 %S A373339 1,1,1,1,4,36,296,2360,19776,180544,1812352,19953792,239490560, %T A373339 3113487872,43589096448,653837077504,10461394714624,177843713556480, %U A373339 3201186851815424,60822550202187776,1216451004083601408,25545471085844758528,562000363888782868480 %N A373339 Number of permutations in symmetric group S_n with an even number of cycles of length 2 or more. %F A373339 a(n) = n!/2 - (n-2)*2^(n-2) = A001710(n) - A036289(n-2). %F A373339 a(n) = A000142(n) - A373340(n). %F A373339 E.g.f.: (1/(1 - x) + exp(2*x)*(1 - x))/2. - _Stefano Spezia_, Jun 05 2024 %e A373339 a(1)=a(2)=a(3)=1 due to S_1,S_2,S_3 containing 1 permutation with an even number of non-fixed point cycles: the identity permutation, with 0 non-fixed point cycles. %e A373339 a(4)=4 due to S_4 containing 4 permutations with an even number of non-fixed point cycles: the 3 (2,2)-cycles (12)(34),(13)(24),(14)(23); and the identity permutation (1)(2)(3)(4). %o A373339 (PARI) a(n) = n!/2 - (n-2)*2^(n-2); \\ _Michel Marcus_, Jun 05 2024 %Y A373339 Cf. A373340 (odd case), A000142, A001710, A036289. %Y A373339 Row sums of triangle A373417. %K A373339 nonn %O A373339 0,5 %A A373339 _Julian Hatfield Iacoponi_, Jun 01 2024