1,3

The sum of each row of the sequence (as a triangular array) is A000272. Second left-downward diagonal is A058877.

Laradji, A. and Umar, A., On the number of nilpotents in the partial symmetric semigroup, Comm. Algebra 32 (2004), 3017-3023.

N(J(n,r)) = C(n,r)*S(n,r+1)*r! where S(n, r + 1) is a Stirling number of the second kind (given by A048993 with zeros removed); generating function = (x+1)^(n-1)

Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 22 2008: (Start)

Define a functional I on formal power series of the form f(x) = 1 + ax + bx^2 + ... by the following iterative process. Define inductively f^(1)(x) = f(x) and f^(n+1)(x) = f(x*f^(n)(x)) for n >= 1. Then set I(f(x)) = lim n -> infinity f^(n)(x) in the x-adic topology on the ring of formal power series; the operator I may also be defined by I(f(x)) := 1/x*series reversion of x/f(x).

Let f(x) = 1 + a*x + a*x^2/2! + a*x^3/3! + ... . Then the e.g.f. for this table is I(f(x)) = 1 + a*x +(a + 2*a^2)*x^2/2! + (a + 9*a^2 + 6*a^3)*x^3/3! + (a + 28*a^2 + 72*a^3 + 24*a^4)*x^4/4! + ... . Note, if we take f(x) = 1 + a*x + a*x^2 + a*x^3 + ... then I(f(x)) is the o.g.f. of the Narayana triangle A001263. (End)

N(J(4,2))=6*6*2=72

Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 22 2008: (Start)

Triangle begins

n\k|..1.....2.....3.....4.....5.....6

=====================================

.1.|..1

.2.|..1.....2

.3.|..1.....9.....6

.4.|..1....28....72....24

.5.|..1....75...500...600...120

.6.|..1...186..2700..7800..5400...720

...

(End)

Cf. A007318, A048993, A000272, A058877.

Sequence in context: A103876 A133174 A155545 this_sequence A061691 A061356 A141028

Adjacent sequences: A141615 A141616 A141617 this_sequence A141619 A141620 A141621

nonn,tabl

A. Umar (aumarh(AT)squ.edu.om), Aug 23 2008