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A135835
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Lower triangular matrix L with first column (1,2,3,4,...,n,...) and otherwise satisfying L(i,j)=Sum[L(i-j-1,k)*L(j,k), k=1..j], read by rows.
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1
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1, 2, 2, 3, 8, 3, 4, 22, 22, 4, 5, 52, 82, 52, 5, 6, 114, 254, 254, 114, 6, 7, 240, 677, 1000, 677, 240, 7, 8, 494, 1692, 3176, 3176, 1692, 494, 8, 9, 1004, 3972, 9136, 12182, 9136, 3972, 1004, 9, 10, 2026, 9052, 24202, 40564, 40564, 24202, 9052, 2026, 10, 11, 4072
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| The definition is equivalent to requiring that L'=L*Transpose(L), where L' is formed from L by shifting column j upward j-1 rows for all j. If the first column is (1,1,1,1,...,1,...} then the lower triangular matrix contains Pascal's triangle. Column two and one-half of column two are essentially A005803 (second-order Eulerian numbers 2^n-2n) and A000295 (Eulerian numbers 2^n-n-1), respectively. Column three has been recently submitted as A135836.
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REFERENCES
| Alan Edelman and Gilbert Strang, Pascal Matrices, Am. Math. Monthly 111(2004)189-197.
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FORMULA
| L(i,1=L(i,i)=i, otherwise L(i,j)=Sum[L(i-j-1,k)*L(j,k)
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EXAMPLE
| Triangle begins :
1 ;
2, 2 ;
3, 8, 3 ;
4, 22, 22, 4 ;
5, 52, 82, 52, 5 ;
6, 114, 254, 254, 114, 6 ; - DELEHAM Philippe, Oct 10 2011
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CROSSREFS
| Cf. A005803, A000295.
Sequence in context: A153216 A141611 A145596 * A177696 A134574 A141617
Adjacent sequences: A135832 A135833 A135834 * A135836 A135837 A135838
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KEYWORD
| nonn,tabl
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AUTHOR
| John W. Layman (layman(AT)math.vt.edu), Nov 30 2007
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