Revision History for A291137
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Showing entries 1-10
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| A291137
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Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of inverse of k-th cyclotomic polynomial.
(history;
published version)
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#21 by Bruno Berselli at Fri Mar 02 03:25:18 EST 2018
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#20 by Ilya Gutkovskiy at Fri Mar 02 02:31:18 EST 2018
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#19 by Ilya Gutkovskiy at Fri Mar 02 02:29:53 EST 2018
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| EXAMPLE
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Square array begins:
Square array begins: 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, -1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, 1, ...
0, -1, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, ...
0, -1, -1, 1, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, ...
0, -1, 1, -1, 1, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, ...
0, -1, -1, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, ...
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#18 by Ilya Gutkovskiy at Fri Mar 02 02:28:51 EST 2018
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| FORMULA
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G.f. of column k, for k > 1, is 1/Phi(k) = productProduct_{d|k} 1/(1 - x^(k/d))^mu(d), where mu() is the Moebius function A008683.
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| EXAMPLE
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Square array begins:: 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
10, -1, 1, 1, 1, , -1, , -1, 0, -1, 1, , -1, 10, 10, 1, , -1, 0, -1, 1, ...
0, -1, -1, -, 1, 0, -1, 0, 0, 1, -10, 0, 0, 1, -10, 0, -1, 1, 0, 10, ...
0, -1, , -1, 0, -1, 0, 0, -1, 0, 0, -1, 0, 0, 0, 1, 0, 0, 1, ...
0, -1, 1, -1, 1, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, ...
0, -1, 1, -1, 10, 0, -, 1, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, ...
0, -1, -1, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, ...
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| STATUS
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approved
editing
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#17 by M. F. Hasler at Thu Mar 01 12:03:03 EST 2018
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#16 by M. F. Hasler at Thu Mar 01 11:58:51 EST 2018
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#15 by M. F. Hasler at Thu Mar 01 11:58:33 EST 2018
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| FORMULA
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Diagonal equals row 0, T(k,k) = T(0,k) = (-1)^[k=1]. - M. F. Hasler, Mar 01 2018
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| PROG
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(PARI) T(n, k)={k||return(!n); n<k||n%=k; polcoeff(1/(polcyclo(k)+O('x^(n+1+n%=k))), n)} \\ M. F. Hasler, Mar 01 2018
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#14 by M. F. Hasler at Thu Mar 01 11:52:47 EST 2018
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| COMMENTS
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Column k is k-periodic, but also satisfies a recurrence relation of order A000010(k) = degree(Phi(k)), with signature given by coefficients of 1-Phi(k)). - _). - _M. F. Hasler_, Feb 16 2018
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| FORMULA
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G.f. of column k, for k > 1, is 1: Product/Phi(k) = product_{d|k} 1/(1 - x^(k/d))^mu(d), where mu() is the Moebius function ( A008683)..
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| EXAMPLE
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1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, -1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, 1, 1, ...
0, -1, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, ...
0, -1, -1, 1, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, ...
0, -1, 1, -1, 1, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...
0, -1, -1, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, ...
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| EXTENSIONS
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References editedEdited by M. F. Hasler, Feb 16 2018, Mar 01 2018
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#13 by M. F. Hasler at Thu Mar 01 11:39:42 EST 2018
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| PROG
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(PARI) T(n, k)={k||return(!n); n<eulerphi(k)||||n%=eulerphi(k); ; polcoeff(1/(polcyclo(k)+O('x^(n+1))), n)} \\ M. F. Hasler, Mar 01 2018
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| STATUS
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proposed
editing
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#12 by M. F. Hasler at Thu Mar 01 11:34:43 EST 2018
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