OFFSET
1,4
COMMENTS
From Gary W. Adamson, Jan 08 2007: (Start)
Let H be this lower triangular matrix. Then:
H * [1, 2, 3, ...] = 1, 3, 5, 8, 9, 15, ... = A018804,
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^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
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.
FORMULA
Sum_{k=1..n} k * T(n, k) = A018804(n). - Gary W. Adamson, Jan 08 2007
From Werner Schulte, Sep 06 2020: (Start)
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)
EXAMPLE
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;
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 *)
PROG
(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
(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 >;
[A054523(n, k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jun 24 2024
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Apr 09 2000
STATUS
approved