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

1,7

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)

Formula

Theorem 1 of Hoskins-Street gives an explicit formula.

Example

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;
  ...

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 *)

Crossrefs

Row sums are (essentially) A000011.

Sequence in context: A029292 A343662 A152198 * A258280 A159255 A376647

Adjacent sequences: A259339 A259340 A259341 * A259343 A259344 A259345

Keyword

nonn,tabf

Author

N. J. A. Sloane, Jun 27 2015

Status

approved