A089231 Triangular array A066667 or A008297 unsigned and transposed.

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

Offset: 1

Author

Philippe Deléham, Dec 10 2003

Keywords

• easy
• nonn
• tabl

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

Examples

			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;
		

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.

Crossrefs

Cf. A000262 (row sums), A008297, A066667, A144084, row mirror of A105278.

Programs

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 *)

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

Formula

T(n, k) = A001263(n, k)*k!; A001263 = triangle of Narayana.

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

Extensions

StackExchange link added by Felix A. Pahl, Dec 25 2012