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A094718 Array T read by antidiagonals: T(n,k) = number of involutions avoiding 132 and 12...k. 11

%I

%S 0,1,0,1,1,0,1,2,1,0,1,2,2,1,0,1,2,3,4,1,0,1,2,3,5,4,1,0,1,2,3,6,8,8,

%T 1,0,1,2,3,6,9,13,8,1,0,1,2,3,6,10,18,21,16,1,0,1,2,3,6,10,19,27,34,

%U 16,1,0,1,2,3,6,10,20,33,54,55,32,1,0,1,2,3,6,10,20,34,61,81,89,32,1

%N Array T read by antidiagonals: T(n,k) = number of involutions avoiding 132 and 12...k.

%C Also, number of paths along a corridor with width k, starting from one side (from H. Bottomley's comment in A061551).

%C Rows converge to binomial(n,floor(n/2)) (A001405).

%H O. Guibert and T. Mansour, <a href="http://www.mat.univie.ac.at/~slc/wpapers/s48guimans.html">Restricted 132-involutions</a>, Séminaire Lotharingien de Combinatoire, B48a (2002), 23 pp.

%H T. Mansour, <a href="https://arxiv.org/abs/math/0302014">Restricted even permutations and Chebyshev polynomials</a>, arXiv:math/0302014 [math.CO], 2003.

%F G.f. for k-th row: 1/(xU(k, 1/(2x))) * Sum_{j=0..k-1} U(j, 1/(2x)), with U(k, x) the Chebyshev polynomials of second kind. - _N. J. A. Sloane_, Dec 20 2008; corrected by _Jean-François Alcover_, Nov 17 2018

%e Array begins

%e 0 0 0 0 0 0 0 0 0 0 ...

%e 1 1 1 1 1 1 1 1 1 1 ...

%e 1 2 2 4 4 8 8 16 16 32 ...

%e 1 2 3 5 8 13 21 34 55 89 ...

%e 1 2 3 6 9 18 27 54 81 162 ...

%e 1 2 3 6 10 19 33 61 108 197 ...

%e 1 2 3 6 10 20 34 68 116 232 ...

%e 1 2 3 6 10 20 35 69 124 241 ...

%e 1 2 3 6 10 20 35 70 125 250 ...

%e 1 2 3 6 10 20 35 70 126 251 ...

%e ...

%t U = ChebyshevU;

%t m = maxExponent = 14;

%t row[1] = Array[0&, m];

%t row[k_] := 1/(x U[k, 1/(2x)])*Sum[U[j, 1/(2x)], {j, 0, k-1}] + O[x]^m // CoefficientList[#, x]& // Rest;

%t T = Table[row[n], {n, 1, m}];

%t Table[T[[n-k+1, k]], {n, 1, m-1}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Nov 17 2018 *)

%Y Rows 3-8 are A016116, A000045, A038754, A028495, A030436, A061551.

%Y Main diagonal is A014495, antidiagonal sums are in A094719.

%Y Cf. A080934 (permutations).

%K nonn,tabl

%O 1,8

%A _Ralf Stephan_, May 23 2004

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Last modified January 17 04:31 EST 2019. Contains 319207 sequences. (Running on oeis4.)