A082905 Modified Pascal-triangle, read by rows. All C(n,j) binomial coefficients are replaced by C(n/g, j/g), where g = gcd(n,j).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 2, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 3, 2, 3, 6, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 8, 4, 56, 2, 56, 4, 8, 1, 1, 9, 36, 3, 126, 126, 3, 36, 9, 1, 1, 10, 5, 120, 10, 2, 10, 120, 5, 10, 1, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1, 1, 12, 6, 4, 3, 792, 2, 792, 3, 4, 6, 12, 1

Offset: 0

Author

Labos Elemer, Apr 23 2003

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,   1;
  1,  4,  2,   4,   1;
  1,  5, 10,  10,   5,   1;
  1,  6,  3,   2,   3,   6,  1;
  1,  7, 21,  35,  35,  21,  7,   1;
  1,  8,  4,  56,   2,  56,  4,   8, 1;
  1,  9, 36,   3, 126, 126,  3,  36, 9,  1;
  1, 10,  5, 120,  10,   2, 10, 120, 5, 10, 1;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A000005, A007318, A056045.

Programs

GAP

T:= function(n,k)
    if k=0 or k=n then return 1;
    else return Binomial(n/Gcd(n,k), k/Gcd(n,k));
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Aug 30 2019

Mathematica

Flatten[Table[Table[Binomial[n/GCD[n, j], j/GCD[n, j]], {j, 0, n}], {n, 1, 32}], 1]

PARI

T(n,k) = my(g=gcd(n,k)); if (!g, g=1); binomial(n/g, k/g);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", "))); \\ Michel Marcus, Aug 30 2019

Sage

def T(n,k):
    if k==0 or k==n: return 1
    else: return binomial(n/gcd(n,k), k/gcd(n,k))
[[T(n,k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 30 2019

Extensions

More terms from Michel Marcus, Aug 30 2019