A051141 - OEIS

A051141

Triangle read by rows: a(n, m) = S1(n, m)*3^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).

1, -3, 1, 18, -9, 1, -162, 99, -18, 1, 1944, -1350, 315, -30, 1, -29160, 22194, -6075, 765, -45, 1, 524880, -428652, 131544, -19845, 1575, -63, 1, -11022480, 9526572, -3191076, 548289, -52920, 2898, -84, 1, 264539520, -239660208

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OFFSET

1,2

COMMENTS

Previous name was: Generalized Stirling number triangle of first kind.

a(n,m) = R_n^m(a=0,b=3) in the notation of the given reference.

a(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m = Product_{j=0..n-1} (x - 3*j), n >= 1 and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).

This is the signed Stirling1 triangle with diagonals d>=0 (main diagonal d=0) scaled with 3^d.

Exponential Riordan array [1/(1 + 3*x), log(1 + 3*x)/3]. The unsigned triangle is [1/(1 - 3*x), log(1/(1 - 3*x)^(1/3))]. - Paul Barry, Apr 29 2009

Also the Bell transform of the triple factorial numbers A032031 which adds a first column (1, 0, 0 ...) on the left side of the triangle and computes the unsigned values. For the definition of the Bell transform, see A264428. See A004747 for the triple factorial numbers A008544 and A203412 for the triple factorial numbers A007559 as well as A039683 and A132062 for the case of double factorial numbers. - Peter Luschny, Dec 21 2015

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales, Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.

Wolfdieter Lang, First 10 rows.

D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962), 1-77.

FORMULA

a(n, m) = a(n-1, m-1) - 3*(n-1)*a(n-1, m) for n >= m >= 1; a(n, m) = 0 for n < m; a(n, 0) = 0 for n >= 1; a(0, 0) = 1.

E.g.f. for the m-th column of the signed triangle: (log(1 + 3*x)/3)^m/m!.

|a(n,1)| = A032031(n-1). - Peter Luschny, Dec 23 2015

EXAMPLE

Triangle starts:

-3, 1;

18, -9, 1;

-162, 99, -18, 1;

1944, -1350, 315, -30, 1;

-29160, 22194, -6075, 765, -45, 1;

524880, -428652, 131544, -19845, 1575, -63, 1;

---

Row polynomial E(3,x) = 18*x-9*x^2+x^3.

From Paul Barry, Apr 29 2009: (Start)

The unsigned array [1/(1 - 3*x), log(1/(1 - 3*x)^(1/3))] has production matrix

3, 1;

9, 6, 1;

27, 27, 9, 1;

81, 108, 54, 12, 1;

243, 405, 270, 90, 15, 1;

729, 1458, 1215, 540, 135, 18, 1;

...

which is A007318^{3} beheaded (by viewing A007318 as a lower triangular matrix). See the comment above. (End)

MATHEMATICA

a[n_, m_] /; n >= m >= 1 := a[n, m] = a[n-1, m-1] - 3(n-1)*a[n-1, m]; a[n_, m_] /; n < m = 0; a[_, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]][[1 ;; 38]] (* Jean-François Alcover, Jun 01 2011, after formula *)

Table[StirlingS1[n, m]*3^(n - m), {n, 1, 10}, {m, 1, n}]//Flatten (* G. C. Greubel, Oct 24 2017 *)

PROG

(Sage) # uses[bell_transform from A264428]

triplefactorial = lambda n: 3^n*factorial(n)

def A051141_row(n):

trifact = [triplefactorial(k) for k in (0..n)]

return bell_transform(n, trifact)

[A051141_row(n) for n in (0..8)] # Peter Luschny, Dec 21 2015

(PARI) for(n=1, 10, for(m=1, n, print1(stirling(n, m, 1)*3^(n-m), ", "))) \\ G. C. Greubel, Oct 24 2017

CROSSREFS

First (m=1) column sequence is: A032031(n-1).

Row sums (signed triangle): A008544(n-1)*(-1)^(n-1).