This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A185411 #80 Jul 14 2023 17:47:18
%S A185411 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,
%T A185411 3508,358,1,0,64,5664,42960,64240,21120,1086,1,0,128,23488,320064,
%U A185411 900560,660880,118632,3272,1,0,256,95872,2225728,10725184,14713840,6049744,638968,9832,1
%N A185411 A triangular decomposition of the double factorial numbers A001147.
%C A185411 Row sums are A001147. Reversal of A185410. Contains A156919 as submatrix.
%C A185411 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
%C A185411 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
%C A185411 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
%H A185411 G. C. Greubel, <a href="/A185411/b185411.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A185411 Jose Bastidas, Christophe Hohlweg, and Franco Saliola, <a href="https://arxiv.org/abs/2306.15556">The Primitive Eulerian polynomial</a>, arXiv:2306.15556 [math.CO], 2023. See Table 2 p. 18.
%H A185411 Shi-Mei Ma, <a href="http://arxiv.org/abs/1204.4963">A family of two-variable derivative polynomials for tangent and secant</a>, arXiv: 1204.4963v3 [math.CO], 2012.
%H A185411 Shi-Mei Ma, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i1p11">A family of two-variable derivative polynomials for tangent and secant</a>, Elect. J. Combinat. 20 (1) (2013) #P11.
%H A185411 Shi-Mei Ma, T. Mansour, and D. Callan, <a href="http://arxiv.org/abs/1404.0731">Some combinatorial arrays related to the Lotka-Volterra system</a>, arXiv:1404.0731 [math.CO], 2014.
%H A185411 Shi-Mei Ma, T. Mansour and D. G. L. Wang, <a href="http://arxiv.org/abs/1403.0233">Combinatorics of Dumont differential system on the Jacobi elliptic functions</a>, arXiv:1403.0233 [math.CO], 2014.
%H A185411 Shi-Mei Ma, Toufik Mansour, David G.L. Wang, and Yeong-Nan Yeh, <a href="https://www.math.sinica.edu.tw/www/file_upload/mayeh/2018SCM-2016-0688.pdf">Several variants of the Dumont differential system and permutation statistics</a>, Science China Mathematics 60 (2018).
%H A185411 Shi-Mei Ma and Y.-N. Yeh, <a href="http://arxiv.org/abs/1503.06601">Stirling permutations, cycle structures of permutations and perfect matchings</a>, arXiv:1503.06601 [math.CO], 2015.
%F A185411 G.f.: 1/(1-xy/(1-2x/(1-3xy/(1-4x/(1-5xy/(1-6x/(1-7xy/(1- ... (continued fraction).
%F A185411 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
%F A185411 T(n,k) = 2^(n-k)*A211399(n,k). - _Philippe Deléham_, Feb 12 2013
%e A185411 Triangle T(n,k) begins:
%e A185411 1;
%e A185411 0, 1;
%e A185411 0, 2, 1;
%e A185411 0, 4, 10, 1;
%e A185411 0, 8, 60, 36, 1;
%e A185411 0, 16, 296, 516, 116, 1;
%e A185411 0, 32, 1328, 5168, 3508, 358, 1;
%e A185411 0, 64, 5664, 42960, 64240, 21120, 1086, 1;
%e A185411 0, 128, 23488, 320064, 900560, 660880, 118632, 3272, 1;
%e A185411 0, 256, 95872, 2225728, 10725184, 14713840, 6049744, 638968, 9832, 1;
%e A185411 ...
%t A185411 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 *)
%Y A185411 Columns 0-1 give: A000007, A131577.
%Y A185411 Cf. A001147, A185410, A156919 (another version).
%K A185411 nonn,easy,tabl
%O A185411 0,5
%A A185411 _Paul Barry_, Jan 26 2011
%E A185411 Sequence terms corrected by _Paul Barry_, Jan 27 2011