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A054523
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Triangle read by rows: T(n,k) = phi(n/k) if k divides n, T(n,k)=0 otherwise (n >= 1, 1 <= k <= n).
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39
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1, 1, 1, 2, 0, 1, 2, 1, 0, 1, 4, 0, 0, 0, 1, 2, 2, 1, 0, 0, 1, 6, 0, 0, 0, 0, 0, 1, 4, 2, 0, 1, 0, 0, 0, 1, 6, 0, 2, 0, 0, 0, 0, 0, 1, 4, 4, 0, 0, 1, 0, 0, 0, 0, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 2, 2, 2, 0, 1, 0, 0, 0, 0, 0, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 6
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OFFSET
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1,4
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COMMENTS
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Let H be this lower triangular matrix. Then:
H * [1, 2, 3, ...] = 1, 3, 5, 8, 9, 15, ... = A018804,
H * sigma(n) = A038040 = d(n) * n = 1, 4, 6, 12, 10, ... where sigma(n) = A000203,
Row sums are A000027 (corrected by Werner Schulte, Sep 06 2020, see comment of Gary W. Adamson, Aug 03 2008),
H^2 * d(n) = d(n)*n, H^2 = A127192,
The Möbius inversion principle of Richard Dedekind and Joseph Liouville (1857), cf. "Concrete Mathematics", p. 136, is equivalent to the statement that row sums are the row index n. - Gary W. Adamson, Aug 03 2008
The multivariable row polynomials give n times the cycle index for the cyclic group C_n, called Z(C_n) (see the MathWorld link with the Harary reference): n*Z(C_n) = Sum_{k=1..n} T(n,k)*(y_{n/k})^k, n >= 1. E.g., 6*Z(C_6) = 2*(y_6)^1 + 2*(y_3)^2 + 1*(y_2)^3 + 1*(y_1)^6. - Wolfdieter Lang, May 22 2012
This is the number of permutations in the n-th cyclic group which are the product of k disjoint cycles. - Robert A. Beeler, Aug 09 2013
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REFERENCES
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Ronald L. Graham, D. E. Knuth, Oren Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., 1994, p. 136.
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LINKS
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FORMULA
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Sum_{k=1..n} k^2 * T(n,k) = A069097(n) for n > 0. (End)
T(2*n-1, n) = A000007(n-1), n >= 1.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (1 - (-1)^n)*n/2.
Sum_{k=1..floor(n+1)/2} T(n-k+1, k) = A092843(n+1).
Sum_{k=1..n} (k+1)*T(n, k) = A209295(n).
Sum_{k=1..n} k^3 * T(n, k) = A343497(n).
Sum_{k=1..n} k^4 * T(n, k) = A343498(n).
Sum_{k=1..n} k^5 * T(n, k) = A343499(n). (End)
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EXAMPLE
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Triangle begins
1;
1, 1;
2, 0, 1;
2, 1, 0, 1;
4, 0, 0, 0, 1;
2, 2, 1, 0, 0, 1;
6, 0, 0, 0, 0, 0, 1;
4, 2, 0, 1, 0, 0, 0, 1;
6, 0, 2, 0, 0, 0, 0, 0, 1;
4, 4, 0, 0, 1, 0, 0, 0, 0, 1;
10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
4, 2, 2, 2, 0, 1, 0, 0, 0, 0, 0, 1;
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MAPLE
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A054523 := proc(n, k) if n mod k = 0 then numtheory[phi](n/k) ; else 0; end if; end proc: # R. J. Mathar, Apr 11 2011
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MATHEMATICA
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T[n_, k_]:= If[k==n, 1, If[Divisible[n, k], EulerPhi[n/k], 0]];
Table[T[n, k], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Dec 15 2017 *)
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PROG
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(Haskell)
a054523 n k = a054523_tabl !! (n-1) !! (k-1)
a054523_row n = a054523_tabl !! (n-1)
a054523_tabl = map (map (\x -> if x == 0 then 0 else a000010 x)) a126988_tabl
(PARI) for(n=1, 10, for(k=1, n, print1(if(!(n % k), eulerphi(n/k), 0), ", "))) \\ G. C. Greubel, Dec 15 2017
(Magma)
A054523:= func< n, k | k eq n select 1 else (n mod k) eq 0 select EulerPhi(Floor(n/k)) else 0 >;
(SageMath)
if (k==n): return 1
elif (n%k)==0: return euler_phi(int(n//k))
else: return 0
flatten([[A054523(n, k) for k in range(1, n+1)] for n in range(1, 16)]) # G. C. Greubel, Jun 24 2024
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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