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Search: id:A111999
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| A111999 |
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A triangle that converts certain binomials into triangle A008276 (diagonals of signed Stirling1 triangle A008275). |
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+0
6
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| -1, 3, 2, -15, -20, -6, 105, 210, 130, 24, -945, -2520, -2380, -924, -120, 10395, 34650, 44100, 26432, 7308, 720, -135135, -540540, -866250, -705320, -303660, -64224, -5040, 2027025, 9459450, 18288270, 18858840, 11098780, 3678840, 623376, 40320, -34459425, -183783600, -416215800
(list; table; graph; listen)
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OFFSET
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1,2
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COMMENT
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Stirling1(n,n-m)=A008275(n,n-m)= sum(a(m,k)*binomial(n,2*m-k),k=0..m-1).
The unsigned column sequences start with A001147, A000906=2*A000457, 2*|A112000|, 4*|A112001|.
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REFERENCES
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Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 152. Table C_{m, nu}
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LINKS
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W. Lang, First 10 rows.
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FORMULA
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a(m, k)=0 if m<k+1; a(1, 0)=-1; a(m, -1):= 0; a(m, k) = -(2*m-k-1)*(a(m-1, k) + a(m-1, k-1)) else.
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CROSSREFS
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Row sums give A032188(m+1)*(-1)^m, m>=1. Unsigned row sums give A032188(m+1), m>=1.
Cf. A008517 (second-order Eulerian triangle) for a similar formula for |Stirling1(n, n-m)|.
Sequence in context: A141235 A051917 A133932 this_sequence A126323 A084886 A055864
Adjacent sequences: A111996 A111997 A111998 this_sequence A112000 A112001 A112002
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KEYWORD
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sign,easy,tabl
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Sep 12 2005
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