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Search: id:A074586
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| A074586 |
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Triangle of Moebius polynomial coefficients, read by rows, the n-th row forming the polynomial M(n,x) such that M(n,-1) = mu(n), the Moebius function of n. |
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9
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| 1, 1, 2, 1, 4, 2, 1, 7, 8, 2, 1, 9, 15, 10, 2, 1, 13, 30, 27, 12, 2, 1, 15, 43, 57, 39, 14, 2, 1, 19, 67, 108, 98, 53, 16, 2, 1, 22, 90, 177, 206, 151, 69, 18, 2, 1, 26, 123, 282, 393, 359, 220, 87, 20, 2, 1, 28, 149, 405, 675, 752, 579, 307, 107, 22, 2, 1, 34, 203, 594, 1109
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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The first few Moebius polynomials are as follows: M(1,x)=1; M(2,x)=1 + 2x; M(3,x)=1 + 4x + 2x^2; M(4,x)=1 + 7x + 8x^2 + 2x^3; M(5,x)=1 + 9x + 15x^2 + 10x^3 + 2x^4; M(6,x)=1 + 13x +30x^2 +27x^3 + 12x^4 + 2x^5; M(7,x)=1 + 15x +43x^2 +57x^3 +39x^4 + 14x^5 + 2x^6; ...
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LINKS
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T. D. Noe, Rows n=0..50 of triangle, flattened
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FORMULA
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The n-th row consists of the coefficients of M(n, x) as a polynomial in x, where M(n, x) = 1 + x*M(1, x)[n/1] + x*M(2, x)[n/2] + x*M(3, x)[n/3] +... + x*M(n-1, x)[n/n-1]; M(1, x) = 1; where [x] = floor(x).
T(n, k) = Sum_{m=1..n-1} [n/m]*T(m, k-1) for n>=k>1, with T(n, 1)=1 for n>=1.
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EXAMPLE
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M(1,x)=1, M(2,x)=1 + 2xM(1,x) = 1 + 2x, M(3,x)=1 + 3xM(1,x) + [3/2]xM(2,x) = 1 + 3x + x(1+2x) = 1 + 4x + 2x^2.
Triangle begins:
1;
1,2;
1,4,2;
1,7,8,2;
1,9,15,10,2;
1,13,30,27,12,2;
1,15,43,57,39,14,2;
1,19,67,108,98,53,16,2;
1,22,90,177,206,151,69,18,2; ...
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PROGRAM
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(PARI) T(n, k)=if(k==1, 1, sum(m=1, n-1, floor(n/m)*T(m, k-1)))
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CROSSREFS
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Cf. A074587.
Sequence in context: A165899 A104582 A133938 this_sequence A134586 A135287 A089606
Adjacent sequences: A074583 A074584 A074585 this_sequence A074587 A074588 A074589
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KEYWORD
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easy,nice,nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Aug 25 2002
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