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Search: id:A101908
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| A101908 |
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Triangle read by rows: Characteristic polynomials of lower triangular Bell number matrix. |
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+0
2
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| 1, -1, 1, -3, 2, 1, -8, 17, -10, 1, -23, 137, -265, 150, 1, -75, 1333, -7389, 13930, -7800, 1, -278, 16558, -277988, 1513897, -2835590, 1583400, 1, -1155, 260364, -14799354, 245309373, -1330523259, 2488395830, -1388641800, 1, -5295, 5042064, -1092706314, 61514634933, -1016911327479
(list; table; graph; listen)
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OFFSET
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1,4
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COMMENT
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Roots of the polynomials are the Bell numbers (A000110) except the leading term.
Second column of the triangle = A024716(n) (partial sums of Bell numbers).
Generation of the triangle: n-th row polynomials are the characteristic polynomial of the lower triangular matrix of the first n rows of the Bell triangle.
So from triangle
1
1 2
2 3 5
5 7 10 15
...
we get characteristic polynomials
x - 1
x^2 - 3*x + 2
x^3 - 8*x^2 + 17*x - 10
x^4 - 23*x^3 + 137*x^2 - 265*x + 150
...
All polynomials (except the first) evaluated at 2 give zero.
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EXAMPLE
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The characteristic polynomial of the 3X3 matrix
1 0 0
1 2 0
2 3 5
= x^3 - 8x^2 + 17x - 10, with roots (1, 2, 5).
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PROGRAM
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(PARI) BM(n) = M=matrix(n, n); M[1, 1]=1; if(n>1, M[2, 1]=1; M[2, 2]=2); \ for(l=3, n, M[l, 1]=M[l-1, l-1]; for(k=2, l, M[l, k]=M[l, k-1]+M[l-1, k-1])); M for(i=1, 10, print(charpoly(BM(i)))) for(i=1, 10, print(round(real(polroots(charpoly(BM(i)))))))
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CROSSREFS
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Cf. A000110, A024716.
Sequence in context: A016648 A104552 A101413 this_sequence A086963 A079749 A002350
Adjacent sequences: A101905 A101906 A101907 this_sequence A101909 A101910 A101911
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KEYWORD
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sign,tabl
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AUTHOR
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Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 28 2005
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