A185411 A triangular decomposition of the double factorial numbers A001147.

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

Offset: 0

Author

Paul Barry, Jan 26 2011

Keywords

• nonn
• easy
• tabl

Comments

Row sums are A001147. Reversal of A185410. Contains A156919 as submatrix.

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 n-1 and, if they were not originally matched to each other, match up their now-unmatched 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

Examples

			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;
  ...
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• Jose Bastidas, Christophe Hohlweg, and Franco Saliola, The Primitive Eulerian polynomial, arXiv:2306.15556 [math.CO], 2023. See Table 2 p. 18.
• Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, arXiv: 1204.4963v3 [math.CO], 2012.
• Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, Elect. J. Combinat. 20 (1) (2013) #P11.
• Shi-Mei Ma, T. Mansour, and D. Callan, Some combinatorial arrays related to the Lotka-Volterra system, arXiv:1404.0731 [math.CO], 2014.
• Shi-Mei Ma, T. Mansour and D. G. L. Wang, Combinatorics of Dumont differential system on the Jacobi elliptic functions, arXiv:1403.0233 [math.CO], 2014.
• Shi-Mei Ma, Toufik Mansour, David G.L. Wang, and Yeong-Nan Yeh, Several variants of the Dumont differential system and permutation statistics, Science China Mathematics 60 (2018).
• Shi-Mei Ma and Y.-N. Yeh, Stirling permutations, cycle structures of permutations and perfect matchings, arXiv:1503.06601 [math.CO], 2015.

Crossrefs

Columns 0-1 give: A000007, A131577.

Cf. A001147, A185410, A156919 (another version).

Programs

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_] /; 1David Callan, Dec 13 2011 *)

Formula

G.f.: 1/(1-xy/(1-2x/(1-3xy/(1-4x/(1-5xy/(1-6x/(1-7xy/(1- ... (continued fraction).

T(n,k) = (2n-2k+1)*T(n-1,k-1) + 2k*T(n-1,k), T(0,0) = 1, T(n,k) = 0 if k < 0 or k > n. - Philippe Deléham, Feb 12 2013

T(n,k) = 2^(n-k)*A211399(n,k). - Philippe Deléham, Feb 12 2013

Extensions

Sequence terms corrected by Paul Barry, Jan 27 2011