OFFSET
1,3
COMMENTS
Row sums: A000262.
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
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
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
For another context for this array see the Callan (2008) article. - Ron L.J. van den Burg, Dec 12 2021
REFERENCES
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 203.
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
David Callan, Sets, Lists and Noncrossing Partitions, Journal of Integer Sequences, Vol. 11 (2008), Article 08.1.3. Also on arXiv, arXiv:0711.4841 [math.CO], 2007-2008.
Tom Copeland, Lagrange a la Lah, 2011.
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015
Olexandr Ganyushkin and Volodymyr Mazorchuk, Combinatorics of nilpotents in symmetric inverse semigroups, Ann. Comb. 8 (2004), no. 2, 161--175. [From Abdullahi Umar, Sep 14 2008]
F. Hivert, J.-C. Novelli and J.-Y. Thibon, Commutative combinatorial Hopf algebras, arXiv:math/0605262 [math.CO], 2006.
Matthieu Josuat-Vergès, Stammering tableaux - Tableaux bégayants, arXiv:1601.02212 [math.CO], 2016. See Lemma 7.1 p. 16.
A. Laradji and A. Umar, On the number of nilpotents in the partial symmetric semigroup, Comm. Algebra 32 (2004), 3017-3023.
Jair Taylor, Number of acyclic digraphs on [n] with k edges and each indegree, outdegree <=1 (question on StackExchange)
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.
FORMULA
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.
From Derek Orr, Mar 12 2015: (Start)
Each row represents a polynomial:
P(1,x) = 1;
P(2,x) = 1 + 2x;
P(3,x) = 1 + 6x + 6x^2;
P(4,x) = 1 + 12x + 36x^2 + 24x^3;
...
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.
(End)
From Peter Bala, Jul 04 2016: (Start)
Working with an offset of 0:
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.
The row polynomials R(n,t) satisfy R(n,t + u) = Sum_{k = 0..n} T(n,k)*t^k*R(n-k,u).
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)
From Peter Bala, Oct 05 2019: (Start)
The following formulas use a column index k starting at 0:
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! + ....
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.
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)
T(n,k) = A105278(n,n-k). - Ron L.J. van den Burg, Dec 12 2021
EXAMPLE
1;
1, 2;
1, 6, 6;
1, 12, 36, 24;
1, 20, 120, 240, 120;
1, 30, 300, 1200, 1800, 720;
1, 42, 630, 4200, 12600, 15120, 5040;
1, 56, 1176, 11760, 58800, 141120, 141120, 40320;
1, 72, 2016, 28224, 211680, 846720, 1693440, 1451520, 362880;
MAPLE
P := n -> simplify(hypergeom([-n, -n+1], [], 1/t));
seq(print(seq(coeff(expand(t^k*P(k)), t, k-j+1), j=1..k)), k=1..n); # Peter Luschny, Oct 29 2014
MATHEMATICA
Table[(Binomial[n - 1, k - 1] Binomial[n, k - 1]/k) k!, {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Jul 04 2016 *)
PROG
(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
CROSSREFS
KEYWORD
AUTHOR
Philippe Deléham, Dec 10 2003
EXTENSIONS
StackExchange link added by Felix A. Pahl, Dec 25 2012
STATUS
approved