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A049403 A triangle of numbers related to triangle A030528. 10
1, 1, 1, 0, 3, 1, 0, 3, 6, 1, 0, 0, 15, 10, 1, 0, 0, 15, 45, 15, 1, 0, 0, 0, 105, 105, 21, 1, 0, 0, 0, 105, 420, 210, 28, 1, 0, 0, 0, 0, 945, 1260, 378, 36, 1, 0, 0, 0, 0, 945, 4725, 3150, 630, 45, 1, 0, 0, 0, 0, 0, 10395, 17325, 6930, 990, 55, 1, 0, 0, 0, 0, 0, 10395, 62370 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

a(n,1) = A019590(n) = A008279(1,n). a(n,m) =: S1(-1; 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))*A001497(n-1,m-1) (signed Bessel triangle). The monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m, E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).

Exponential Riordan array [1+x, x(1+x/2)]. T(n,k) = A001498(k+1, n-k). - Paul Barry, Jan 15 2009

LINKS

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

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

W. Lang, First 10 rows of the array and more. [From Wolfdieter Lang, Oct 17 2008]

FORMULA

a(n, m) = n!*A030528(n, m)/(m!*2^(n-m)); a(n, m) = (2*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: ((x*(1+x/2))^m)/m!.

a(n,m) = A122848(n,m). - R. J. Mathar, Jan 14 2011

EXAMPLE

{1};

{1,1};

{0,3,1}; row polynomial E(3,x)= 3*x^2 + x^3.

{0,3,6,1};

...

MAPLE

# The function BellMatrix is defined in A264428.

# Adds (1, 0, 0, 0, ..) as column 0.

BellMatrix(n -> `if`(n<2, 1, 0), 9); # Peter Luschny, Jan 28 2016

MATHEMATICA

t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); Table[ t[n, k], {n, 11}, {k, n}] // Flatten

CROSSREFS

Cf. A000085 (row sums).

Sequence in context: A255123 A244483 A292727 * A104556 A116089 A122016

Adjacent sequences:  A049400 A049401 A049402 * A049404 A049405 A049406

KEYWORD

easy,nonn,tabl

AUTHOR

Wolfdieter Lang

STATUS

approved

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Last modified February 21 04:43 EST 2018. Contains 299389 sequences. (Running on oeis4.)