

A185411


A triangular decomposition of the double factorial numbers A001147.


6



1, 0, 1, 0, 2, 1, 0, 4, 10, 1, 0, 8, 60, 36, 1, 0, 16, 296, 516, 116, 1, 0, 32, 1328, 5168, 3508, 358, 1, 0, 64, 5664, 42960, 64240, 21120, 1086, 1, 0, 128, 23488, 320064, 900560, 660880, 118632, 3272, 1, 0, 256, 95872, 2225728, 10725184, 14713840, 6049744, 638968, 9832, 1
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OFFSET

0,5


COMMENTS

Row n counts perfect matchings of [2n] by number of matches in which the smaller entry is odd. For example, T(2,1)=2 counts 13/24, 14/23, in each of which only the first matching pair has an odd smaller entry. Outline proof. Consider the map on perfect matchings of [2n] given by "delete the entries n and n1 and, if they were not originally matched to each other, match up their nowunmatched partners". Consideration of this map and its effect on the statistic "number of matches in which the smaller entry is odd" yields the Mathematica recurrence below.  David Callan, Dec 13 2011
Triangle T(n,k), 0 <= k <= n, given by (0, 2, 0, 4, 0, 6, 0, 8, 0, 10, 0, ...) DELTA (1, 0, 3, 0, 5, 0, 7, 0, 9, 11, 0, ...) where DELTA is the operator defined in A084938.  Philippe Deléham, Feb 12 2013
T(n,k), 0 <= k <= n, is the number of signed permutations of [n] that are products of balanced cycles (i.e., cuspidal elements of the type B Coxeter group) and have excedance number of type B equal to k.  Jose Bastidas, Jul 05 2023


LINKS



FORMULA

G.f.: 1/(1xy/(12x/(13xy/(14x/(15xy/(16x/(17xy/(1 ... (continued fraction).
T(n,k) = (2n2k+1)*T(n1,k1) + 2k*T(n1,k), T(0,0) = 1, T(n,k) = 0 if k < 0 or k > n.  Philippe Deléham, Feb 12 2013


EXAMPLE

Triangle T(n,k) begins:
1;
0, 1;
0, 2, 1;
0, 4, 10, 1;
0, 8, 60, 36, 1;
0, 16, 296, 516, 116, 1;
0, 32, 1328, 5168, 3508, 358, 1;
0, 64, 5664, 42960, 64240, 21120, 1086, 1;
0, 128, 23488, 320064, 900560, 660880, 118632, 3272, 1;
0, 256, 95872, 2225728, 10725184, 14713840, 6049744, 638968, 9832, 1;
...


MATHEMATICA

u[n_, 0] := If[n==0, 1, 0]; u[n_, m_] /; m==1 := 2^(n  1); u[n_, m_] /; m==n>=1 := 1; u[n_, m_] /; 1<m<n := u[n, m] = (2m)*u[n  1, m] + (2n  2m + 1)*u[n  1, m  1]; Flatten[Table[u[n, m], {n, 0, 10}, {m, 0, n}]] (* David Callan, Dec 13 2011 *)


CROSSREFS



KEYWORD



AUTHOR



EXTENSIONS

Sequence terms corrected by Paul Barry, Jan 27 2011


STATUS

approved



