A092186
a(n) = 2(m!)^2 for n = 2m and m!(m+1)! for n = 2m+1.
Original entry on oeis.org
2, 1, 2, 2, 8, 12, 72, 144, 1152, 2880, 28800, 86400, 1036800, 3628800, 50803200, 203212800, 3251404800, 14631321600, 263363788800, 1316818944000, 26336378880000, 144850083840000, 3186701844480000, 19120211066880000, 458885065605120000, 2982752926433280000
Offset: 0
- A. O. Munagi, Alternating subsets and permutations, Rocky Mountain J. Math. 40 (6) (2010) 1965-1977 doi:10.1216/RJM-2010-40-6-1965, Corollary 3.2.
- David Singmaster, Problem 1654, Mathematics Magazine 75 (October 2002). Solution in Mathematics Magazine 76 (October 2003).
-
a:= proc(n) option remember; `if`(n<2, 2-n,
(n*(3*n-1)*(n-1)*a(n-2) -4*a(n-1))/(12*n-16))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Nov 11 2013
-
f[n_] := If[EvenQ[n], 2 (n/2)!^2, ((n + 1)/2)! ((n - 1)/2)!]; Table[
f[n], {n, 0, 25}] (* Geoffrey Critzer, Aug 24 2013 *)
A047677
Row 2 of square array defined in A047675: 2*n!*(n+1)!.
Original entry on oeis.org
2, 4, 24, 288, 5760, 172800, 7257600, 406425600, 29262643200, 2633637888000, 289700167680000, 38240422133760000, 5965505852866560000, 1085722065221713920000, 228001633696559923200000, 54720392087174381568000000, 14883946647711431786496000000
Offset: 0
-
a:= proc(n) a(n):= `if`(n=0, 2, n*(n+1) * a(n-1)) end:
seq(a(n), n=0..20); # Alois P. Heinz, Nov 11 2013
-
2*Times@@@Partition[Range[0,20]!,2,1] (* Harvey P. Dale, Sep 25 2017 *)
A152876
Number of permutations of {1,2,...,n} having no consecutive triples of the form (odd, even, odd) or (even, odd, even).
Original entry on oeis.org
1, 1, 2, 4, 16, 60, 288, 1584, 10368, 74880, 604800, 5356800, 51840000, 544320000, 6147187200, 74579097600, 962415820800, 13241346048000, 192255565824000, 2957575348224000, 47721518530560000, 811595019755520000, 14407079038894080000, 268390402745794560000
Offset: 0
a(3) = 4 because we have 132, 213, 231 and 312.
-
ae := proc (n) options operator, arrow: 2*(sum((n^2-3*n*j+3*j^2)*binomial(n-j, j)^2/(n-j)^2, j = 0 .. floor((1/2)*n))) end proc: ao := proc (n) options operator, arrow: sum(binomial(n+1-k, k)*binomial(n-k, k)*(2*n^2+2*n-6*k*n-3*k+6*k^2)/((n+1-k)*(n-k)), k = 0 .. floor((1/2)*n)) end proc: a := proc (n) if `mod`(n, 2) = 0 then factorial((1/2)*n)^2*ae((1/2)*n) else factorial((1/2)*n-1/2)*factorial((1/2)*n+1/2)*ao((1/2)*n-1/2) end if end proc: 1, 1, seq(a(n), n = 2 .. 22);
# second Maple program:
b:= proc(o, u, t) option remember; `if`(u+o=0, 1,
`if`(t=4, 0, o*b(o-1, u, `if`(t=3, 5, 2)))+
`if`(t=5, 0, u*b(o, u-1, `if`(t=2, 4, 3))))
end:
a:= n-> b(ceil(n/2), floor(n/2), 1):
seq(a(n), n=0..30); # Alois P. Heinz, Nov 11 2013
-
b[o_, u_, t_] := b[o, u, t] = If[u+o == 0, 1, If[t==4, 0, o*b[o-1, u, If[t==3, 5, 2]]] + If[t==5, 0, u*b[o, u-1, If[t==2, 4, 3]]]]; a[n_] := b[Ceiling[n/2], Floor[n/2], 1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 29 2015, after Alois P. Heinz *)
A329550
Total number of consecutive triples of the form (odd, even, odd) or (even, odd, even) in all permutations of [n].
Original entry on oeis.org
0, 0, 0, 2, 16, 108, 864, 7200, 69120, 705600, 8064000, 97977600, 1306368000, 18441561600, 281652940800, 4533271142400, 78111748915200, 1412288317440000, 27115935694848000, 544201764986880000, 11524272670310400000, 254238259854458880000, 5887622859787468800000
Offset: 0
-
a:= proc(n) option remember; `if`(n<5, [0$3, 2, 16][n+1],
(n-2)*(2*(n-4)*a(n-1)+(n-3)^2*n*a(n-2))/(n-3)/(n-4))
end:
seq(a(n), n=0..30);
-
a[n_] := a[n] = If[n < 5, {0, 0, 0, 2, 16}[[n+1]],
(n-2)*(2*(n-4)*a[n-1] + (n-3)^2*n*a[n-2])/(n-3)/(n-4)];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 21 2022, after Alois P. Heinz *)
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