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A049404
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Triangle read by rows, the Bell transform of n!*binomial(2,n) (without column 0).
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8
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1, 2, 1, 2, 6, 1, 0, 20, 12, 1, 0, 40, 80, 20, 1, 0, 40, 360, 220, 30, 1, 0, 0, 1120, 1680, 490, 42, 1, 0, 0, 2240, 9520, 5600, 952, 56, 1, 0, 0, 2240, 40320, 48720, 15120, 1680, 72, 1, 0, 0, 0, 123200, 332640, 184800, 35280, 2760, 90, 1, 0, 0, 0, 246400, 1786400
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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 A049324.
a(n,1) = A008279(2,n-1). a(n,m) =: S1(-2; 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))*A004747(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).
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LINKS
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FORMULA
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a(n, m) = n!*A049324(n, m)/(m!*3^(n-m));
a(n, m) = (3*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+x^2+(x^3)/3)^m)/m!.
a(n,m) = n!/(3^m * m!)*(Sum_{i=0..floor(m-n/3)} (-1)^i * binomial(m,i) * binomial(3*m-3*i,n)), 0 for empty sums. - Werner Schulte, Feb 20 2020
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EXAMPLE
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E.g. row polynomial E(3,x) = 2*x+6*x^2+x^3.
Triangle starts:
{1}
{2, 1}
{2, 6, 1}
{0, 20, 12, 1}
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MATHEMATICA
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rows = 11;
a[n_, m_] := BellY[n, m, Table[k! Binomial[2, 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(2, 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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