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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).
3
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
0,5
EXAMPLE
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;
MATHEMATICA
Flatten[Table[Table[Binomial[n/GCD[n, j], j/GCD[n, j]], {j, 0, n}], {n, 1, 32}], 1]
PROG
(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
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Labos Elemer, Apr 23 2003
EXTENSIONS
More terms from Michel Marcus, Aug 30 2019
STATUS
approved