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A049411
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Triangle read by rows, the Bell transform of n!*binomial(5,n) (without column 0).
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3
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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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!.
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EXAMPLE
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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}
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MATHEMATICA
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rows = 10;
a[n_, m_] := BellY[n, m, Table[k! Binomial[5, k], {k, 0, rows}]];
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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