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A049403
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A triangle of numbers related to triangle A030528.
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6
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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; internal format)
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OFFSET
| 1,5
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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(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).
Exponential Riordan array [1+x,x(1+x/2)]. T(n,k)=A001498(k+1,n-k). [From Paul Barry (pbarry(AT)wit.ie), Jan 15 2009]
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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!*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
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EXAMPLE
| {1}; {1,1}; {0,3,1}; {0,3,6,1}; ... E.g. row polynomial E(3,x)= 3*x^2+x^3.
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MATHEMATICA
| t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); Table[ t[n, k], {n, 11}, {k, n}] // Flatten
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CROSSREFS
| Cf. A000085 (row sums).
Sequence in context: A099546 A036870 A036874 * A104556 A116089 A122016
Adjacent sequences: A049400 A049401 A049402 * A049404 A049405 A049406
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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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