A199542 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).

1, 0, 1, -12, 0, 1, 0, -96, 0, 1, -320, 0, -400, 0, 1, 0, 8640, 0, -1200, 0, 1, -53760, 0, 188160, 0, -2940, 0, 1, 0, 4300800, 0, 1630720, 0, -6272, 0, 1, -27869184, 0, 3870720, 0, 9144576, 0, -12096, 0, 1, 0, 4877107200, 0, -1548288000, 0, 38949120, 0

Offset

1,4

Comments

Triangle T(n,m)*m!^2/n!^2=

1. Riordan Array (1,x^2*cotan(x)) without first column.

2. Riordan Array (x*cotan(x),x^2*cotan(x)) numbering triangle (0,0).

Links

Table of n, a(n) for n=1..52.

D. Kruchinin and V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, Journal of Integer Sequence, Vol. 15 (2012), article 12.9.3.

V. V. Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Dmitry V. Kruchinin and Vladimir V. Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6.

T. Mansour, M. Shattuck and D. G. L. Wang, Counting subwords in flattened permutations, arXiv preprint arXiv:1307.3637 [math.CO], 2013.

Formula

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)!)))).

Example

T(n,m)*m!^2/n!^2=
1
0, 1
-1/3, 0, 1
0, -2/3, 0, 1
-1/45, 0, -1, 0, 1
0, 1/15, 0, -4/3, 0, 1
-2/945, 0, 4/15, 0, -5/3, 0, 1

Mathematica

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 *)

Prog

(Maxima)
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);
(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)!))));
{for(n=1, 10, for(m=1, n, print1(T(n, m), ", "); ); print(); ); } \\ Indranil Ghosh, Mar 10 2017

Crossrefs

Sequence in context: A230526 A156401 A246223 * A304330 A322731 A370330

Adjacent sequences: A199539 A199540 A199541 * A199543 A199544 A199545

Keyword

sign,tabl

Author

Vladimir Kruchinin, Nov 07 2011

Status

approved