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Search: id:A155517
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| A155517 |
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Triangle read by rows: T(n,k) is the number of permutations p of {1,2,...,n} for which the number of j < ceil(n/2) such that p(j)+p(n+1-j)=n+1 is equal to k (n>=1; 0<=k <=ceil(n/2)). [For the permutation 31756284 of S_8 we have k=2 because p(2)+p(7)=1+8=9 and p(3)+p(6)=7+2=9; for the permutation 3214756 of S_7 we have k=2 because p(3)+p(5)=1+7=8 and p(4)+p(4)=4+4=8.] |
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+0
3
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| 0, 1, 0, 2, 4, 0, 2, 16, 0, 8, 64, 48, 0, 8, 384, 288, 0, 48, 2880, 1536, 576, 0, 48, 23040, 12288, 4608, 0, 384, 208896, 115200, 30720, 7680, 0, 384, 2088960, 1152000, 307200, 76800, 0, 3840, 23193600, 12533760, 3456000, 614400, 115200, 0, 3840, 278323200
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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Row sums are the factorial numbers (A000142).
Row n contains 1+ceil(n/2)entries.
T(2n,n)=n!*2^n=A037223(2n)=number of centrosymmetric permutations in S[2n];
T(2n+1,n+1)=n!*2^n=A037223(2n+1)=number of centrosymmetric permutations in S[2n+1].
T(n,0)=A155518(n).
Sum(k*T(n,k),k=0..ceil(n/2))=A155519(n).
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FORMULA
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T(2n,k)=n!*2^n*A055140(n,k);
T(2n-1,k)=(n-1)!*2^(n-1)*A055140(n,k);
here A055140(n,k)=A053871(n-k)*C(n,k), where g(n)=A053871(n) is defined by g(0)=1, g(1)=0, g(n)=2(n-1)[g(n-1)+g(n-2)].
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EXAMPLE
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T(4,2)=8 because we have 1234, 4231, 1324, 4321, 2143, 3142, 2413 and 3412.
Triangle starts:
0,1;
0,2;
4,0,2;
16,0,8;
64,48,0,8;
384,288,0,48;
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MAPLE
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g[0] := 1: g[1] := 0: for n from 2 to 20 do g[n] := (2*(n-1))*(g[n-1]+g[n-2]) end do: T := proc (n, k) if `mod`(n, 2) = 0 then 2^((1/2)*n)*factorial((1/2)*n)*g[(1/2)*n-k]*binomial((1/2)*n, k) else 2^((1/2)*n-1/2)*factorial((1/2)*n-1/2)*g[(1/2)*n+1/2-k]*binomial((1/2)*n+1/2, k) end if end proc: for n to 12 do seq(T(n, k), k = 0 .. ceil((1/2)*n)) end do;
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CROSSREFS
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A000142, A037223, A155518, A155519, A055140
Sequence in context: A153182 A111818 A007631 this_sequence A123514 A064178 A018220
Adjacent sequences: A155514 A155515 A155516 this_sequence A155518 A155519 A155520
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 26 2009
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