A054523 Triangle read by rows: T(n,k) = phi(n/k) if k divides n, T(n,k)=0 otherwise (n >= 1, 1 <= k <= n).

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

Offset: 1

Author

N. J. A. Sloane, Apr 09 2000

Keywords

• nonn
• tabl

Comments

From Gary W. Adamson, Jan 08 2007: (Start)

Let H be this lower triangular matrix. Then:

H * A051731 = A126988,

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,

H * d(n) (A000005) = 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,

H * mu(n) (A008683) = A007431(n) (corrected by Werner Schulte, Sep 06 2020),

H^2 row sums = A018804. (End)

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

See A102190 (no 0's, rows reversed). - Wolfdieter Lang, May 29 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

Examples

			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;
		

References

• Ronald L. Graham, D. E. Knuth, Oren Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., 1994, p. 136.

Links

• Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
• Eric Weisstein's World of Mathematics, Cycle Index.

Crossrefs

Cf. A000005, A000007, A000010, A000012, A000203, A007431, A008683.

Cf. A038040, A051731, A054521, A054525, A102190, A126988.

Sums incliude: A029935, A069097, A092843 (diagonal), A209295.

Sums of the form Sum_{k} k^p * T(n, k): A000027 (p=0), A018804 (p=1), A069097 (p=2), A343497 (p=3), A343498 (p=4), A343499 (p=5).

Programs

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
-- Reinhard Zumkeller, Jan 20 2014

Magma

A054523:= func< n,k | k eq n select 1 else (n mod k) eq 0 select EulerPhi(Floor(n/k)) else 0 >;
[A054523(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jun 24 2024

Maple

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

Mathematica

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 *)

PARI

for(n=1, 10, for(k=1, n, print1(if(!(n % k), eulerphi(n/k), 0), ", "))) \\ G. C. Greubel, Dec 15 2017

SageMath

def A054523(n,k):
    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

Formula

Sum_{k=1..n} k * T(n, k) = A018804(n). - Gary W. Adamson, Jan 08 2007

Equals A054525 * A126988 as infinite lower triangular matrices. - Gary W. Adamson, Aug 03 2008

From Werner Schulte, Sep 06 2020: (Start)

Sum_{k=1..n} T(n,k) * A000010(k) = A029935(n) for n > 0.

Sum_{k=1..n} k^2 * T(n,k) = A069097(n) for n > 0. (End)

From G. C. Greubel, Jun 24 2024: (Start)

T(2*n-1, n) = A000007(n-1), n >= 1.

T(2*n, n) = A000012(n), 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)