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%I #42 Mar 10 2017 10:27:47
%S 1,0,1,-12,0,1,0,-96,0,1,-320,0,-400,0,1,0,8640,0,-1200,0,1,-53760,0,
%T 188160,0,-2940,0,1,0,4300800,0,1630720,0,-6272,0,1,-27869184,0,
%U 3870720,0,9144576,0,-12096,0,1,0,4877107200,0,-1548288000,0,38949120,0
%N Triangle T(n,m) = coefficient of x^n in expansion of (x^2*cotan(x))^m = sum(n>=m, T(n,m) x^n * m!^2/n!^2).
%C Triangle T(n,m)*m!^2/n!^2=
%C 1. Riordan Array (1,x^2*cotan(x)) without first column.
%C 2. Riordan Array (x*cotan(x),x^2*cotan(x)) numbering triangle (0,0).
%H D. Kruchinin and V. Kruchinin, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Kruchinin/kruchinin5.html">A Method for Obtaining Generating Function for Central Coefficients of Triangles</a>, Journal of Integer Sequence, Vol. 15 (2012), article 12.9.3.
%H V. V. Kruchinin, D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%H Dmitry V. Kruchinin and Vladimir V. Kruchinin, <a href="http://www.emis.de/journals/JIS/VOL18/Kruchinin/kruch9.pdf">A Generating Function for the Diagonal T_{2n,n} in Triangles</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6.
%H T. Mansour, M. Shattuck and D. G. L. Wang, <a href="http://arxiv.org/abs/1307.3637">Counting subwords in flattened permutations</a>, arXiv preprint arXiv:1307.3637 [math.CO], 2013.
%F T(n,m) = n!^2/m!^2*2^(n-2*m)*(-1)^((n-m)/2)*sum(l=0..m, (2^l*l!*binomial(m,l)* sum(k..0,n-2*m+l,(k!*stirling1(l+k,l)*stirling2(n-2*m+l,k))/((l+k)!*(n-2*m+l)!)))).
%e T(n,m)*m!^2/n!^2=
%e 1
%e 0, 1
%e -1/3, 0, 1
%e 0, -2/3, 0, 1
%e -1/45, 0, -1, 0, 1
%e 0, 1/15, 0, -4/3, 0, 1
%e -2/945, 0, 4/15, 0, -5/3, 0, 1
%t Table[n!^2/(m!)^2*2^(n - 2 m) (-1)^((n - m)/2) Sum[(2^l (l!) Binomial[m, l] Sum[(k! StirlingS1[l + k, l] StirlingS2[n - 2 m + l, k])/((l + k)! (n - 2 m + l)!), {k, 0, n - 2 m + l}]), {l, 0, m}], {n, 10}, {m, n}] // Flatten (* _Michael De Vlieger_, Apr 26 2016 *)
%o (Maxima)
%o T(n,m):=n!^2/m!^2*2^(n-2*m)*(-1)^((n-m)/2)*sum((2^l*l!*binomial(m,l)* sum((k!*stirling1(l+k,l)*stirling2(n-2*m+l,k))/((l+k)!*(n-2*m+l)!),k,0,n-2*m+l)),l,0,m);
%o (PARI) T(n,m) = n!^2/m!^2*2^(n - 2*m)*(-1)^((n - m)/2)*sum(l=0, m, (2^l*l!*binomial(m, l)*sum(k=0, n - 2*m + l, (k!*stirling(l + k, l, 1)*stirling(n - 2*m + l, k, 2))/((l + k)!*(n - 2*m + l)!))));
%o {for(n=1, 10, for(m=1, n, print1(T(n, m),", ");); print();); } \\ _Indranil Ghosh_, Mar 10 2017
%K sign,tabl
%O 1,4
%A _Vladimir Kruchinin_, Nov 07 2011
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