login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A049411 Triangle read by rows, the Bell transform of n!*binomial(5,n) (without column 0). 3
1, 5, 1, 20, 15, 1, 60, 155, 30, 1, 120, 1300, 575, 50, 1, 120, 9220, 8775, 1525, 75, 1, 0, 55440, 114520, 36225, 3325, 105, 1, 0, 277200, 1315160, 730345, 112700, 6370, 140, 1, 0, 1108800, 13428800, 13000680, 3209745, 291060, 11130, 180, 1, 0, 3326400 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Previous name was: A triangle of numbers related to triangle A049327.

a(n,1) = A008279(5,n-1). a(n,m) =: S1(-5; n,m), a member of a sequence of lower triangular Jabotinsky matrices, including S1(1; n,m) = A008275 (signed Stirling first kind), S1(2; n,m) = A008297(n,m) (signed Lah numbers). a(n,m) matrix is inverse to signed matrix ((-1)^(n-m))*A013988(n,m).

The monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).

For the definition of the Bell transform see A264428 and the link. - Peter Luschny, Jan 16 2016

LINKS

Table of n, a(n) for n=1..47.

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Peter Luschny, The Bell transform

FORMULA

a(n, m) = n!*A049327(n, m)/(m!*6^(n-m));

a(n, m) = (6*m-n+1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1;

a(n, m) = 0, n<m; a(n, 0) := 0; a(1, 1) = 1.

E.g.f. for m-th column: (((-1+(1+x)^6)/6)^m)/m!.

EXAMPLE

Row polynomial E(3,x) = 20*x + 15*x^2 + x^3.

Triangle starts:

{  1}

{  5,    1}

{ 20,   15,   1}

{ 60,  155,  30,  1}

{120, 1300, 575, 50, 1}

MATHEMATICA

rows = 10;

a[n_, m_] := BellY[n, m, Table[k! Binomial[5, k], {k, 0, rows}]];

Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-Fran├žois Alcover, Jun 22 2018 *)

PROG

(Sage) # uses[bell_matrix from A264428]

# Adds 1, 0, 0, 0, ... as column 0 at the left side of the triangle.

bell_matrix(lambda n: factorial(n)*binomial(5, n), 8) # Peter Luschny, Jan 16 2016

CROSSREFS

Cf. A049327.

Row sums give A049428.

Sequence in context: A088577 A127561 A144879 * A070729 A296306 A101693

Adjacent sequences:  A049408 A049409 A049410 * A049412 A049413 A049414

KEYWORD

easy,nonn,tabl

AUTHOR

Wolfdieter Lang

EXTENSIONS

New name from Peter Luschny, Jan 16 2016

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
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 6 12:30 EDT 2020. Contains 336246 sequences. (Running on oeis4.)