A259342 Irregular triangle read by rows: T(n,k) = number of equivalence classes of binary sequences of length n containing exactly 2k ones.

1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 4, 1, 1, 4, 8, 4, 1, 1, 4, 10, 7, 1, 1, 5, 16, 16, 5, 1, 1, 5, 20, 26, 10, 1, 1, 6, 29, 50, 29, 6, 1, 1, 6, 35, 76, 57, 14, 1, 1, 7, 47, 126, 126, 47, 7, 1, 1, 7, 56, 185, 232, 111, 19, 1, 1, 8, 72, 280, 440, 280, 72, 8, 1

Offset: 1

Author

N. J. A. Sloane, Jun 27 2015

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  1;
  1, 1;
  1, 1;
  1, 2,  1;
  1, 2,  1;
  1, 3,  3,   1;
  1, 3,  4,   1;
  1, 4,  8,   4,   1;
  1, 4, 10,   7,   1;
  1, 5, 16,  16,   5,  1;
  1, 5, 20,  26,  10,  1;
  1, 6, 29,  50,  29,  6, 1;
  1, 6, 35,  76,  57, 14, 1;
  1, 7, 47, 126, 126, 47, 7, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..200, flattened
• W. D. Hoskins and Anne Penfold Street, Twills on a given number of harnesses, J. Austral. Math. Soc. Ser. A 33 (1982), no. 1, 1-15.
• W. D. Hoskins and A. P. Street, Twills on a given number of harnesses, J. Austral. Math. Soc. (Series A), 33 (1982), 1-15. (Annotated scanned copy)

Crossrefs

Row sums are (essentially) A000011.

Programs

Maple

with(numtheory):
T:= (n, k)-> (add(`if`(irem(2*k*d, n)=0, phi(n/d)
     *binomial(d, 2*k*d/n), 0), d=divisors(n))
     +n*binomial(iquo(n, 2), k))/(2*n):
seq(seq(T(n, k), k=0..n/2), n=1..20);  # Alois P. Heinz, Jun 28 2015

Mathematica

T[n_, k_] := (DivisorSum[n, If[Mod[2k*#, n]==0, EulerPhi[n/#]*Binomial[#, 2k*#/n], 0]&] + n*Binomial[Quotient[n, 2], k])/(2n); Table[T[n, k], {n, 1, 20}, { k, 0, n/2}] // Flatten (* Jean-François Alcover, Feb 28 2017, after Alois P. Heinz *)

Formula

Theorem 1 of Hoskins-Street gives an explicit formula.