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A089231 Triangular array A066667 or A008297 unsigned and transposed.

%I A089231 #92 Jan 18 2022 14:11:27
%S A089231 1,1,2,1,6,6,1,12,36,24,1,20,120,240,120,1,30,300,1200,1800,720,1,42,
%T A089231 630,4200,12600,15120,5040,1,56,1176,11760,58800,141120,141120,40320,
%U A089231 1,72,2016,28224,211680,846720,1693440,1451520,362880
%N A089231 Triangular array A066667 or A008297 unsigned and transposed.
%C A089231 Row sums: A000262.
%C A089231 T(n, k) is also the number of nilpotent partial one-one bijections (of an n-element set) of height k (height(alpha) = |Im(alpha)|). - _Abdullahi Umar_, Sep 14 2008
%C A089231 T(n, k) is also the number of acyclic directed graphs on n labeled nodes with k-1 edges with all indegrees and outdegrees at most 1. - _Felix A. Pahl_, Dec 25 2012
%C A089231 For n > 1, the n-th derivative of exp(1/x) is of the form (exp(1/x)/x^(2*n))*(P(n-1,x)) where P(n-1,x) is a polynomial of degree n-1 with n terms. The term of degree k in P(n-1,x) has a coefficient given by T(n-1,k). Example: The third derivative of exp(1/x) is (exp(1/x)/x^6)*(1+6x+6x^2) and the 3rd row of this triangle is 1, 6, 6, which corresponds to this coefficients of the polynomial 1+6x+6x^2. - _Derek Orr_, Nov 06 2014
%C A089231 For another context for this array see the Callan (2008) article. -  _Ron L.J. van den Burg_, Dec 12 2021
%D A089231 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 203.
%H A089231 G. C. Greubel, <a href="/A089231/b089231.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A089231 David Callan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Callan/callan412.html">Sets, Lists and Noncrossing Partitions</a>, Journal of Integer Sequences, Vol. 11 (2008), Article 08.1.3. Also <a href="http://arxiv.org/abs/0711.4841">on arXiv</a>, arXiv:0711.4841 [math.CO], 2007-2008.
%H A089231 Tom Copeland, <a href="https://tcjpn.wordpress.com/2011/04/11/lagrange-a-la-lah/">Lagrange a la Lah</a>, 2011.
%H A089231 Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/12/21/generators-inversion-and-matrix-binomial-and-integral-transforms/">Generators, Inversion, and Matrix, Binomial, and Integral Transforms</a>, 2015
%H A089231 Olexandr Ganyushkin and Volodymyr Mazorchuk, <a href="http://dx.doi.org/10.1007/s00026-004-0213-7">Combinatorics of nilpotents in symmetric inverse semigroups</a>, Ann. Comb. 8 (2004), no. 2, 161--175. [From _Abdullahi Umar_, Sep 14 2008]
%H A089231 F. Hivert, J.-C. Novelli and J.-Y. Thibon, <a href="https://arxiv.org/abs/math/0605262">Commutative combinatorial Hopf algebras</a>, arXiv:math/0605262 [math.CO], 2006.
%H A089231 Matthieu Josuat-Vergès, <a href="http://arxiv.org/abs/1601.02212">Stammering tableaux - Tableaux bégayants</a>, arXiv:1601.02212 [math.CO], 2016. See Lemma 7.1 p. 16.
%H A089231 A. Laradji and A. Umar, <a href="http://dx.doi.org/10.1081/AGB-120038637">On the number of nilpotents in the partial symmetric semigroup</a>, Comm. Algebra 32 (2004), 3017-3023.
%H A089231 Jair Taylor, <a href="http://math.stackexchange.com/questions/263945">Number of acyclic digraphs on [n] with k edges and each indegree, outdegree <=1</a> (question on StackExchange)
%H A089231 Jian Zhou, <a href="https://arxiv.org/abs/2108.10514">On Some Mathematics Related to the Interpolating Statistics</a>, arXiv:2108.10514 [math-ph], 2021.
%F A089231 T(n, k) = A001263(n, k)*k!; A001263 = triangle of Narayana.
%F A089231 T(n, k) = C(n, n-k+1)*(n-1)!/(n-k)! = Sum_{i=n-k+1..n} |S1(n, i)*S2(i, n-k+1)| , with S1, S2 the Stirling numbers.
%F A089231 From _Derek Orr_, Mar 12 2015: (Start)
%F A089231 Each row represents a polynomial:
%F A089231 P(1,x) = 1;
%F A089231 P(2,x) = 1 + 2x;
%F A089231 P(3,x) = 1 + 6x + 6x^2;
%F A089231 P(4,x) = 1 + 12x + 36x^2 + 24x^3;
%F A089231 ...
%F A089231 They are related through P(n+1,x) = x^2*P'(n,x) - (1+2*n*x)*P(n,x) with P(1,x) = 1.
%F A089231 (End)
%F A089231 From _Peter Bala_, Jul 04 2016: (Start)
%F A089231 Working with an offset of 0:
%F A089231 G.f.: exp(x*t)*I_1(2*sqrt(x)) = 1 + (1 + 2*t)*x/(1!*2!) + (1 + 6*t + 6*t^2)*x^2/(2!*3!) + (1 + 12*t + 36*t^2 + 24*t^3)*x^3/(3!*4!) + ..., where I_1(x) = Sum_{n >= 0} (x/2)^(2*n)/(n!*(n+1)!) is a modified Bessel function of the first kind.
%F A089231 The row polynomials R(n,t) satisfy R(n,t + u) = Sum_{k = 0..n} T(n,k)*t^k*R(n-k,u).
%F A089231 R(n,t) = 1 + Sum_{k = 0..n-1} (-1)^(n-k+1)*(n+1)!/(k+1)!* binomial(n,k)*t^(n-k)*R(k,t). Cf. A144084. (End)
%F A089231 From _Peter Bala_, Oct 05 2019: (Start)
%F A089231 The following formulas use a column index k starting at 0:
%F A089231 E.g.f.: exp(x/(1 - t*x)) - 1 = x + (1 + 2*t)*x^2/2! + (1 + 6*t + 6*t^2)*x^3/3! + ....
%F A089231 Recurrence for row polynomials: R(n+1,t) = (1 + 2*n*t)R(n,t) - n*(n-1)*t^2*R(n-1,t), with R(1,t) = 1 and R(2,t) = 1 + 2*t.
%F A089231 R(n+1,t) equals the numerator polynomial of the finite continued fraction 1 + n*t/(1 + n*t/(1 + (n-1)*t/(1 + (n-1)*t/(1 + ... + 2*t/(1 + 2*t/(1 + t/(1 + t/(1)))))))). The denominator polynomial is the n-th row polynomial of A144084. (End)
%F A089231 T(n,k) = A105278(n,n-k). - _Ron L.J. van den Burg_, Dec 12 2021
%e A089231 1;
%e A089231 1,  2;
%e A089231 1,  6,    6;
%e A089231 1, 12,   36,    24;
%e A089231 1, 20,  120,   240,    120;
%e A089231 1, 30,  300,  1200,   1800,    720;
%e A089231 1, 42,  630,  4200,  12600,  15120,    5040;
%e A089231 1, 56, 1176, 11760,  58800, 141120,  141120,   40320;
%e A089231 1, 72, 2016, 28224, 211680, 846720, 1693440, 1451520, 362880;
%p A089231 P := n -> simplify(hypergeom([-n,-n+1],[],1/t));
%p A089231 seq(print(seq(coeff(expand(t^k*P(k)),t,k-j+1),j=1..k)),k=1..n); # _Peter Luschny_, Oct 29 2014
%t A089231 Table[(Binomial[n - 1, k - 1] Binomial[n, k - 1]/k) k!, {n, 9}, {k, n}] // Flatten (* _Michael De Vlieger_, Jul 04 2016 *)
%o A089231 (PARI) tabl(nn) = {for (n=0, nn, for (k=0, n, print1((n+1)!*binomial(n,k)/(n-k+1)!, ", ");); print(););} \\ _Michel Marcus_, Jan 12 2016
%Y A089231 Cf. A000262 (row sums), A008297, A066667, A144084, row mirror of A105278.
%K A089231 easy,nonn,tabl
%O A089231 1,3
%A A089231 _Philippe Deléham_, Dec 10 2003
%E A089231 StackExchange link added by _Felix A. Pahl_, Dec 25 2012