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 A137782 a(n) = the number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) where, for each k (2<=k<=n), the sign of (p(k) - p(k-1)) equals the sign of (p(n+2-k) - p(n+1-k)). 3
 1, 1, 2, 2, 12, 24, 200, 540, 6160, 21616, 306432, 1310880, 22338624, 113017696, 2245983168, 13108918368, 297761967360, 1969736890624, 50332737128960, 372125016868608, 10565549532009472, 86337114225206784, 2696451226217269248, 24132714802787013632 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Number of order n permutations whose descent set is invariant w.r.t. the function f(x)=n-x. - Max Alekseyev, May 06 2009 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..190 FORMULA a(2n) = A000984(n)*A060350(n). - Max Alekseyev, Apr 23 2009 EXAMPLE Consider the permutation (for n = 7): 3,6,7,5,1,2,4 The signs of the differences between adjacent terms forms the sequence: ++--++, which has reflective symmetry. So this permutation, among others, is counted when n = 7. MAPLE b:= proc(u, o, h) option remember; `if`(u+o=0, 1,       add(add(b(u-j, o+j-1, h+i-1), i=1..u+o-h), j=1..u)+       add(add(b(u+j-1, o-j, h-i), i=1..h), j=1..o))     end: a:= proc(n) option remember; local r;      `if`(irem(n, 2, 'r')=0, b(0, r\$2)*binomial(n, r),       add(add(binomial(j-1, i)*binomial(n-j, r-i)*       b(r-i, i, n-j-r+i), i=0..min(j-1, r)), j=1..n))     end: seq(a(n), n=0..30);  # Alois P. Heinz, Sep 15 2015 PROG (PARI) { a(n) = local(r, u, c, t); r=0; forvec(v=vector(n-1, i, [2*i==n, 1]), u=sum(i=1, #v, v[i]); c=sum(i=1, (n-1)\2, !v[i]&&!v[n-i]); t=[0]; for(i=1, #v, if(v[i], t=concat(t, [i]))); r += (-1)^u * 2^c * n! \ prod(i=2, #t, (t[i]-t[i-1])!) \ (n-t[ #t])! ); (-1)^(n+1)*r } [From Max Alekseyev, May 06 2009] CROSSREFS Cf. A137783. Sequence in context: A092900 A164961 A122007 * A131384 A052612 A287951 Adjacent sequences:  A137779 A137780 A137781 * A137783 A137784 A137785 KEYWORD nonn AUTHOR Leroy Quet, Feb 10 2008 EXTENSIONS First 8 terms calculated by Olivier Gérard. Extended by Max Alekseyev, May 06 2009 a(0), a(22) from Alois P. Heinz, Jul 02 2015 a(23) from Alois P. Heinz, Sep 15 2015 STATUS approved

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