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A013988 Triangle read by rows, the inverse Bell transform of n!*binomial(5,n) (without column 0). 9
1, 5, 1, 55, 15, 1, 935, 295, 30, 1, 21505, 7425, 925, 50, 1, 623645, 229405, 32400, 2225, 75, 1, 21827575, 8423415, 1298605, 103600, 4550, 105, 1, 894930575, 358764175, 59069010, 5235405, 271950, 8330, 140, 1, 42061737025, 17398082625 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Previous name was: Triangle of numbers related to triangle A049224; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.

a(n,m) := S2p(-5; n,m), a member of a sequence of triangles including S2p(-1; n,m) := A001497(n-1,m-1) (Bessel triangle) and ((-1)^(n-m))*S2p(1; n,m) := A008277(n,m) (Stirling 2nd kind). a(n,1)= A008543(n-1).

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..38.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.

M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3

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

Peter Luschny, The Bell transform

Index entries for sequences related to Bessel functions or polynomials

FORMULA

a(n, m) = n!*A049224(n, m)/(m!*6^(n-m));

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

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

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

EXAMPLE

{    1}

{    5,    1}

{   55,   15,   1}

{  935,  295,  30,  1}

{21505, 7425, 925, 50, 1}

MATHEMATICA

rows = 10;

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

A = Table[b[n, m], {n, 1, rows}, {m, 1, rows}] // Inverse // Abs;

A013988 = Table[A[[n, m]], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-Fran├žois Alcover, Jun 22 2018 *)

PROG

(Sage)

# The function inverse_bell_matrix is defined in A264428.

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

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

CROSSREFS

Cf. A001497, A008277, A049224.

Cf. A000369, A004747, A011801, A028844.

Sequence in context: A144341 A144342 A144268 * A246006 A050970 A138548

Adjacent sequences:  A013985 A013986 A013987 * A013989 A013990 A013991

KEYWORD

easy,nonn,tabl

AUTHOR

Wolfdieter Lang

EXTENSIONS

New name from Peter Luschny, Jan 16 2016

STATUS

approved

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Last modified December 13 22:55 EST 2019. Contains 329974 sequences. (Running on oeis4.)