login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A132393 Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows. 71
1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 6, 11, 6, 1, 0, 24, 50, 35, 10, 1, 0, 120, 274, 225, 85, 15, 1, 0, 720, 1764, 1624, 735, 175, 21, 1, 0, 5040, 13068, 13132, 6769, 1960, 322, 28, 1, 0, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1, 0, 362880, 1026576, 1172700 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Triangle T(n,k), 0<=k<=n, read by rows given by [0,1,1,2,2,3,3,4,4,5,5,...] DELTA [1,0,1,0,1,0,1,0,1,...] where DELTA is the operator defined in A084938.

A094645*A007318 as infinite lower triangular matrices.

Row sums are the factorial numbers. - Roger L. Bagula, Apr 18 2008

Exponential Riordan array [1/(1-x), log(1/(1-x))]. - Ralf Stephan, Feb 07 2014

Also the Bell transform of the factorial numbers (A000142). For the definition of the Bell transform see A264428 and for cross-references A265606. - Peter Luschny, Dec 31 2015

This is the lower triagonal Sheffer matrix of the associated or Jabotinsky type |S1| = (1, -log(1-x)) (see the W. Lang link under A006232 for the notation and references). This implies the e.g.f.s given below. |S1| is the transition matrix from the monomial basis {x^n} to the rising factorial basis {risefac(x,n)}, n >= 0. - Wolfdieter Lang, Feb 21 2017

REFERENCES

Roland Bacher, P De La Harpe, Conjugacy growth series of some infinitely generated groups. 2016. hal-01285685v2; https://hal.archives-ouvertes.fr/hal-01285685/document

Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 149-150

LINKS

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

J. Fernando Barbero G., Jesús Salas, Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013

Ricky X. F. Chen, A Note on the Generating Function for the Stirling Numbers of the First Kind, Journal of Integer Sequences, 18 (2015), #15.3.8.

FindStat - Combinatorial Statistic Finder, The number of saliances of a permutation, The number of cycles in the cycle decomposition of a permutation.

W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939.

John M. Holte, Carries, Combinatorics and an Amazing Matrix, The American Mathematical Monthly, Vol. 104, No. 2 (Feb., 1997), pp. 138-149.

Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO]. - From N. J. A. Sloane, Aug 21 2012

X.-T. Su, D.-Y. Yang, W.-W. Zhang, A note on the generalized factorial, Australasian Journal of Combinatorics, Volume 56 (2013), Pages 133-137.

FORMULA

T(n,k) = T(n-1,k-1)+(n-1)*T(n-1,k), n,k>=1; T(n,0)=T(0,k); T(0,0)=1 .

Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A000142(n), A001147(n), A007559(n), A007696(n), A008548(n), A008542(n), A045754(n), A045755(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively. - Philippe Deléham, Nov 13 2007

Expand 1/(1-t)^x = Sum[p(x,n)*t^n/n!,{n,0,Infinity}]; then the coefficients of the p(x,n) produce the triangle. - Roger L. Bagula, Apr 18 2008

Sum_{k=0..n} T(n,k)*2^k*x^(n-k) = A000142(n+1), A000165(n), A008544(n), A001813(n), A047055(n), A047657(n), A084947(n), A084948(n), A084949(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Sep 18 2008

a(n) = Sum_{k=0..n} T(n,k)*3^k*x^(n-k) = A001710(n+2), A001147(n+1), A032031(n), A008545(n), A047056(n), A011781(n), A144739(n), A144756(n), A144758(n) for x=1,2,3,4,5,6,7,8,9,respectively. - Philippe Deléham, Sep 20 2008

Sum_{k=0..n} T(n,k)*4^k*x^(n-k) = A001715(n+3), A002866(n+1), A007559(n+1), A047053(n), A008546(n), A049308(n), A144827(n), A144828(n), A144829(n) for x=1,2,3,4,5,6,7,8,9 respectively. - Philippe Deléham, Sep 21 2008

Sum_{k=0..n} x^k*T(n,k) = x*(1+x)*(2+x)*...*(n-1+x), n>=1. - Philippe Deléham, Oct 17 2008

From Wolfdieter Lang, Feb 21 2017:

E.g.f. m-th column: (-log(1 - x))^m, m >= 0.

E.g.f. triangle (see the Apr 18 2008 Baluga comment): exp(-x*ln(1-z)).

E.g.f. a-sequence: x/(1 - exp(-x)). See A164555/A027642. The e.g.f. for the z-sequence is 0. (End)

EXAMPLE

The triangle T(n,k) begins:

n\k 0     1      2      3     4     5    6   7  8 9

0:  1

1:  0     1

2:  0     1      1

3:  0     2      3      1

4:  0     6     11      6    1

5:  0    24    274    225    85    15    1

7:  0   720   1764   1624   735   175   21   1

8:  0  5040  13068  13132  6769  1960  322  28  1

9:  0 40320 109584 118124 67284 22449 4536 546 36 1

...

row n =10: 0 362880 1026576 1172700 723680 269325 63273 9450 870 45 1.

... reformatted - Wolfdieter Lang, Feb 21 2017

---------------------------------------------------

Production matrix is

0, 1

0, 1, 1

0, 1, 2, 1

0, 1, 3, 3, 1

0, 1, 4, 6, 4, 1

0, 1, 5, 10, 10, 5, 1

0, 1, 6, 15, 20, 15, 6, 1

0, 1, 7, 21, 35, 35, 21, 7, 1

...

MAPLE

a132393_row := proc(n) local k; seq(coeff(expand(pochhammer (x, n)), x, k), k=0..n) end: # Peter Luschny, Nov 28 2010

MATHEMATICA

p[t_] = 1/(1 - t)^x; Table[ ExpandAll[(n!)SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[(n!)* CoefficientList[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a] (* Roger L. Bagula, Apr 18 2008 *)

Flatten[Table[Abs[StirlingS1[n, i]], {n, 0, 10}, {i, 0, n}]] (* Harvey P. Dale, Feb 04 2014 *)

PROG

(Maxima) create_list(abs(stirling1(n, k)), n, 0, 12, k, 0, n); /* Emanuele Munarini, Mar 11 2011 */

(Haskell)

a132393 n k = a132393_tabl !! n !! k

a132393_row n = a132393_tabl !! n

a132393_tabl = map (map abs) a048994_tabl

-- Reinhard Zumkeller, Nov 06 2013

CROSSREFS

Essentially a duplicate of A048994. Cf. A008275, A008277, A130534.

Sequence in context: A264430 A264433 A048994 * A121434 A137329 A265604

Adjacent sequences:  A132390 A132391 A132392 * A132394 A132395 A132396

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Nov 10 2007, Oct 15 2008, Oct 17 2008

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified April 23 18:45 EDT 2017. Contains 285329 sequences.