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A089231 Triangular array A066667 or A008297 unsigned and transposed. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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

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

T. Copeland, Lagrange a la Lah , 2011.

T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015

Olexandr Ganyushkin, 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)

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)

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

Cf. A000262 (row sums), A008297, A066667, A144084.

Sequence in context: A046651 A063007 A202190 * A052296 A019538 A269646

Adjacent sequences:  A089228 A089229 A089230 * A089232 A089233 A089234

KEYWORD

easy,nonn,tabl

AUTHOR

Philippe Deléham, Dec 10 2003

EXTENSIONS

Comment and link added by Felix A. Pahl, Dec 25 2012

STATUS

approved

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Last modified February 21 16:38 EST 2018. Contains 299414 sequences. (Running on oeis4.)