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A049404
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A triangle of numbers related to triangle A049324.
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5
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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
| 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 (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 17 2008]
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FORMULA
| 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!.
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EXAMPLE
| {1}; {2,1}; {2,6,1}; {0,20,12,1}; ... E.g. row polynomial E(3,x)= 2*x+6*x^2+x^3.
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CROSSREFS
| Row sums give A049425.
Sequence in context: A006602 A144824 A144358 * A159885 A178803 A083773
Adjacent sequences: A049401 A049402 A049403 * A049405 A049406 A049407
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
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