A271187 Triangle T(n,k) read by rows: T(n,k) is the squarefree part of C(n,k).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 1, 6, 1, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 5, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 2, 7, 14, 70, 14, 7, 2, 1, 1, 1, 1, 21, 14, 14, 21, 1, 1, 1, 1, 10, 5, 30, 210, 7, 210, 30, 5, 10, 1, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1, 1, 3, 66, 55, 55, 22, 231, 22, 55, 55, 66, 3, 1, 1, 13, 78, 286, 715, 143, 429, 429, 143, 715, 286, 78, 13, 1

Offset

0,5

Links

Table of n, a(n) for n=0..104.

Example

Triangle starts:
[1]
[1, 1]
[1, 2, 1]
[1, 3, 3, 1]
[1, 1, 6, 1, 1]
[1, 5, 10, 10, 5, 1]
[1, 6, 15, 5, 15, 6, 1]
[1, 7, 21, 35, 35, 21, 7, 1]
[1, 2, 7, 14, 70, 14, 7, 2, 1]
[1, 1, 1, 21, 14, 14, 21, 1, 1, 1]
[1, 10, 5, 30, 210, 7, 210, 30, 5, 10, 1]
[1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1]
[1, 3, 66, 55, 55, 22, 231, 22, 55, 55, 66, 3, 1]
[1, 13, 78, 286, 715, 143, 429, 429, 143, 715, 286, 78, 13, 1]
[1, 14, 91, 91, 1001, 2002, 3003, 858, 3003, 2002, 1001, 91, 91, 14, 1]
[1, 15, 105, 455, 1365, 3003, 5005, 715, 715, 5005, 3003, 1365, 455, 105, 15, 1]
...

Maple

T:= (n, k)-> mul(i[1]^irem(i[2], 2),
     i=ifactors(binomial(n, k))[2]):
seq(seq(T(n, k), k=0..n), n=0..15);  # Alois P. Heinz, Apr 01 2016

Mathematica

Table[Times @@ Apply[Power, {First@ #, Mod[Last@ #, 2]} & /@ FactorInteger@ Binomial[n, k], {1}], {n, 0, 13}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 01 2016, after Zak Seidov at A007913 *)

Prog

(PARI) for(n=0, 20, for(k=0, n, print1(core(binomial(n, k)), ", ")));
(Magma) /* As triangle: */ [[Squarefree(Binomial(n, k)): k in [0..n]]: n in [0..20]]; // Wesley Ivan Hurt, Apr 01 2016
(Sage) [squarefree_part(binomial(n, k)) for n in (0..15) for k in (0..n) ] # Bruno Berselli, Apr 01 2016

Crossrefs

Cf. A007318 (binomial coefficients), A007913 (squarefree parts).

T(2n,n) gives A069113.

Sequence in context: A117185 A129181 A157694 * A093557 A098802 A048804

Adjacent sequences: A271184 A271185 A271186 * A271188 A271189 A271190

Keyword

nonn,tabl

Author

Joerg Arndt, Apr 01 2016

Status

approved