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 A373340 #23 Jun 28 2024 22:52:55 %S A373340 0,0,1,5,20,84,424,2680,20544,182336,1816448,19963008,239511040, %T A373340 3113532928,43589194752,653837290496,10461395173376,177843714539520, %U A373340 3201186853912576,60822550206644224,1216451004093038592,25545471085864681472,562000363888824811520 %N A373340 Number of permutations of symmetric group S_n with an odd number of cycles of length 2 or more. %F A373340 a(n) = n!/2 + (n-2)*2^(n-2) = A001710(n) + A036289(n-2). %F A373340 a(n) = A000142(n) - A373339(n). %F A373340 E.g.f.: (1/(1 - x) - exp(2*x)*(1 - x))/2. - _Stefano Spezia_, Jun 05 2024 %e A373340 a(0)=0 due to the sole permutation in S_0 being the empty permutation, with 0 non-fixed point cycles, not an odd number. %e A373340 a(1)=0 due to the sole permutation in S_1 being the fixed point (1), with 0 non-fixed point cycles, not an odd number. %e A373340 a(2)=1 due to 1 permutation in S_2 with an odd number of non-fixed point cycles: (12), with 1 non-fixed point cycle. %e A373340 a(3)=5 due to 5 permutations in S_3 with an odd number of non-fixed point cycles: (12)(3),(13)(2),(23)(1),(123),(132), all with 1 non-fixed point cycle. %o A373340 (PARI) a(n) = n!/2 + (n-2)*2^(n-2); \\ _Michel Marcus_, Jun 05 2024 %Y A373340 Cf. A373339 (even case), A000142, A001710, A036289. %Y A373340 Row sums of triangle A373418. %K A373340 nonn %O A373340 0,4 %A A373340 _Julian Hatfield Iacoponi_, Jun 01 2024