A123610 Triangle read by rows, where T(n,k) = (1/n)*Sum_{d|(n,k)} phi(d) * binomial(n/d,k/d)^2 for n >= k > 0, with T(n,0) = 1 for n >= 0.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 10, 4, 1, 1, 5, 20, 20, 5, 1, 1, 6, 39, 68, 39, 6, 1, 1, 7, 63, 175, 175, 63, 7, 1, 1, 8, 100, 392, 618, 392, 100, 8, 1, 1, 9, 144, 786, 1764, 1764, 786, 144, 9, 1, 1, 10, 205, 1440, 4420, 6352, 4420, 1440, 205, 10, 1, 1, 11, 275, 2475, 9900

Offset: 0

Author

Paul D. Hanna, Oct 03 2006

Keywords

• nonn
• tabl

Comments

A variant of the triangle A047996 of circular binomial coefficients.

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,  1;
  1,  2,   1;
  1,  3,   3,    1;
  1,  4,  10,    4,    1;
  1,  5,  20,   20,    5,    1;
  1,  6,  39,   68,   39,    6,    1;
  1,  7,  63,  175,  175,   63,    7,    1;
  1,  8, 100,  392,  618,  392,  100,    8,   1;
  1,  9, 144,  786, 1764, 1764,  786,  144,   9,  1;
  1, 10, 205, 1440, 4420, 6352, 4420, 1440, 205, 10, 1;
  ...
Example of column g.f.s are:
column 1: 1/(1 - x)^2;
column 2: Ser([1, 1, 3, 1]) / ((1 - x)^2*(1 - x^2)^2) = g.f. of A005997;
column 3: Ser([1, 2, 11, 26, 30, 26, 17, 6, 1]) / ((1 - x)^2*(1 - x^2)^2*(1 -x^3)^2);
column 4: Ser([1, 3, 28, 94, 240, 440, 679, 839, 887, 757, 550, 314, 148, 48, 11, 1]) / ((1 - x)^2*(1 - x^2)^2*(1 - x^3)^2*(1 - x^4)^2);
where Ser() denotes a polynomial in x with the given coefficients, as in Ser([1, 1, 3, 1]) = (1 + x + 3*x^2 + x^3).
		

Links

• Paul D. Hanna, Rows n = 0..45, flattened.
• Petros Hadjicostas, Proofs of some formulae for g.f.'s of this sequence.

Crossrefs

Cf. Columns: A005997, A123613, A123614, A123615, A123616.

Cf. A123611 (row sums), A123612 (antidiagonal sums), A123617 (central terms).

Cf. A123618, A123619, A047996 (variant), A128545.

Programs

Mathematica

T[, 0] = 1; T[n, k_] := 1/n DivisorSum[n, If[GCD[k, #] == #, EulerPhi[#]* Binomial[n/#, k/#]^2, 0]&]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 06 2015, adapted from PARI *)

PARI

{T(n,k)=if(k==0,1,(1/n)*sumdiv(n,d,if(gcd(k,d)==d, eulerphi(d)*binomial(n/d,k/d)^2,0)))}

Formula

T(2*n+1, n) = (2*n + 1)*A000108(n)^2 = (2*n + 1)*((2*n)!/(n!(n+1)!))^2 = A000891(n) for n >= 0.

Row sums are 2*A047996(2*n,n) = 2*A003239(n) for n > 0.

Row sums equal the row sums of triangle A128545.

For n >= 1, the g.f. of column n has the form: P_n(x)/(Product_{m=1..n} (1 - x^m)^2), where P_n(x) is a polynomial with n^2 coefficients such that the sum of the coefficients is P_n(1) = (2*n - 1)!.

From Petros Hadjicostas, Oct 24 2017: (Start)

Proofs of the following formulae can be found in the links.

G.f.: Sum_{n>=1, k>=0} T(n,k)*x^n*y^k = -Sum_{s>=1} (phi(s)/s)*log(f(x^s,y^s)), where phi(s) is Euler's totient function at s, f(x,y) = (sqrt(g(x,y)) + 1 -(1 + y)*x)/2, and g(x,y) = 1 - 2*(1 + y)*x + (1 - y)^2*x^2. (Term T(0,0) is not used in this g.f.)

Row g.f.: Sum_{k>=0} T(n,k)*y^k = (1/n)*Sum_{d|n} phi(d)*R(n/d, y^d), where R(m, y) = [z^m] (1 + (1 + y)*z + y*z^2)^m. (End)