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A049410 A triangle of numbers related to triangle A049325. 5
1, 3, 1, 6, 9, 1, 6, 51, 18, 1, 0, 210, 195, 30, 1, 0, 630, 1575, 525, 45, 1, 0, 1260, 10080, 6825, 1155, 63, 1, 0, 1260, 51660, 71505, 21840, 2226, 84, 1, 0, 0, 207900, 623700, 333585, 57456, 3906, 108, 1, 0, 0, 623700, 4573800, 4293135, 1195425, 131670 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n,1)= A008279(3,n-1). a(n,m)=: S1(-3; 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))*A000369(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).

Also the inverse Bell transform of the quadruple factorial numbers Product_{k=0..n-1} (4*k+3) (A008545) adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015

LINKS

Table of n, a(n) for n=1..52.

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

FORMULA

a(n, m) = n!*A049325(n, m)/(m!*4^(n-m)); a(n, m) = (4*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)^4)/4)^m)/m!.

EXAMPLE

Triangle begins:

  {1};

  {3,1};

  {6,9,1};

  {6,51,18,1};

  ...

E.g. row polynomial E(3,x)= 6*x+9*x^2+x^3.

MATHEMATICA

rows = 10;

t = Table[Product[4k+3, {k, 0, n-1}], {n, 0, rows}];

T[n_, k_] := BellY[n, k, t];

M = Inverse[Array[T, {rows, rows}]] // Abs;

A049325 = Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-Fran├žois Alcover, Jun 22 2018, after Peter Luschny *)

PROG

(Sage) # uses[inverse_bell_transform from A265605]

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

multifact_4_3 = lambda n: prod(4*k + 3 for k in (0..n-1))

inverse_bell_matrix(multifact_4_3, 9) # Peter Luschny, Dec 31 2015

CROSSREFS

Row sums give A049426.

Sequence in context: A257259 A074475 A144877 * A013610 A008573 A089710

Adjacent sequences:  A049407 A049408 A049409 * A049411 A049412 A049413

KEYWORD

easy,nonn,tabl

AUTHOR

Wolfdieter Lang

STATUS

approved

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Last modified May 31 07:30 EDT 2020. Contains 334747 sequences. (Running on oeis4.)