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A132191
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Square array a(m,n) read by antidiagonals, defined by A000010(n)*a(m,n) = Sum_{k=1..n, gcd(k,n)=1} m^{ Sum_{d|n} A000010(d)/ (multiplicative order of k modulo d) }.
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13
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1, 1, 2, 1, 4, 3, 1, 6, 9, 4, 1, 12, 18, 16, 5, 1, 12, 54, 40, 25, 6, 1, 40, 72, 160, 75, 36, 7, 1, 28, 405, 280, 375, 126, 49, 8, 1, 96, 390, 2176, 825, 756, 196, 64, 9, 1, 104, 1944, 2800, 8125, 2016, 1372, 288, 81, 10, 1, 280, 3411, 17920, 13175, 23976, 4312, 2304, 405
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OFFSET
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1,3
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COMMENTS
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Number of step shifted (decimated) sequences of length n using a maximum of m different symbols. See A056371 for an explanation of step shifts. -
Number of mappings with domain {0..n-1} and codomain {1..m} up to equivalence. Mappings A and B are equivalent if there is a d, prime to n, such that A(i) = B(i*d mod n) for i in {0..n-1}. (End)
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LINKS
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EXAMPLE
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Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2, 4, 6, 12, 12, 40, 28, 96, 104, 280, 216, 1248, 704, 2800, 4344, 8928, 8232, 44224, 29204, 136032, ...
3, 9, 18, 54, 72, 405, 390, 1944, 3411, 14985, 17802, 139968, 133104, 798525, 1804518, 5454378, 8072532, 64599849, 64573626, 437732424, ...
4, 16, 40, 160, 280, 2176, 2800, 17920, 44224, 263296, 419872, 4280320, 5594000, 44751616, 134391040, 539054080, 1073758360, 11453771776, 15271054960, 137575813120, ...
5, 25, 75, 375, 825, 8125, 13175, 103125, 327125, 2445625, 4884435, 61640625, 101732425, 1017323125, 3816215625, 19104609375, 47683838325, 635787765625, 1059638680675, 11924780390625, ...
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MATHEMATICA
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a[m_, n_] := (1/EulerPhi[n])*Sum[If[GCD[k, n]==1, m^DivisorSum[n, EulerPhi[#] / MultiplicativeOrder[k, #]&], 0], {k, 1, n}]; Table[a[m-n+1, n], {m, 1, 15}, {n, m, 1, -1}] // Flatten (* Jean-François Alcover, Dec 01 2015 *)
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PROG
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(PARI) for(i=1, 15, for(m=1, i, n=i-m+1; print1(sum(k=1, n, if(gcd(k, n)==1, m^sumdiv(n, d, eulerphi(d)/znorder(Mod(k, d))), 0))/eulerphi(n)", "))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 26 2008
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 26 2008
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STATUS
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approved
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