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A102190
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Triangle read by rows: coefficients of cycle index polynomial for the cyclic group C_n, Z(C_n,x), multiplied by n.
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11
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1, 1, 1, 1, 2, 1, 1, 2, 1, 4, 1, 1, 2, 2, 1, 6, 1, 1, 2, 4, 1, 2, 6, 1, 1, 4, 4, 1, 10, 1, 1, 2, 2, 2, 4, 1, 12, 1, 1, 6, 6, 1, 2, 4, 8, 1, 1, 2, 4, 8, 1, 16, 1, 1, 2, 2, 6, 6, 1, 18, 1, 1, 2, 4, 4, 8, 1, 2, 6, 12, 1, 1, 10, 10, 1, 22, 1, 1, 2, 2, 2, 4, 4, 8, 1, 4, 20, 1, 1, 12, 12, 1, 2, 6, 18, 1, 1, 2, 6, 6
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| Row n gives the coefficients of x[k]^{n/k} with increasing divisors k of n.
The length of row n is tau(n) = A000005(n) (number of divisors of n, including 1 and n).
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REFERENCES
| F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1994; pp. 181 and 184.
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LINKS
| W. Lang, More terms and comments.
Eric Weisstein's World of Mathematics, Cycle Index.
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FORMULA
| a(n, m)= phi(k(m)), m=1..tau(n), n>=1, with k(m) the m-th divisor of n, written in increasing order.
Z(C_n, x):=sum(sum(phi(k)*x[k]^{n/k}, k|n))/n, where phi(n)= A000010(n) (Euler's totient function) and k|n means 'k divides n'. Cf. Harary reference and MathWorld link.
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EXAMPLE
| [1], [1, 1], [1, 2], [1, 1, 2], [1, 4], [1, 1, 2, 2], [1, 6],...
Z(C_6,x)=(1*x[1]^6 + 1*x[2]^3 + 2*x[3]^2 + 2*x[6]^1)/6.
a(6,1)=phi(1)=1, a(6,2)=phi(2)=1, a(6,3)=phi(3)=2, a(6,4)=phi(6)=2.
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MATHEMATICA
| k[n_, m_] := Divisors[n][[m]]; a[n_, m_] := EulerPhi[k[n, m]]; Flatten[Table[a[n, m], {n, 1, 28}, {m, 1, DivisorSigma[0, n]}]] (* From Jean-François Alcover, Jul 25 2011, after given formula *)
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CROSSREFS
| Sequence in context: A033809 A046067 A132066 * A138650 A137843 A130194
Adjacent sequences: A102187 A102188 A102189 * A102191 A102192 A102193
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KEYWORD
| nonn,easy,tabf
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Feb 15 2005
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