OFFSET
1,4
COMMENTS
Row sums = A000041 starting (1, 2, 3, 5, 7, 11, 15, ...).
T(n,k) is the number of partitions of n into parts with GCD = k. - Alois P. Heinz, Jun 06 2013
LINKS
Alois P. Heinz, Rows n = 1..141, flattened
FORMULA
EXAMPLE
First few rows of the triangle:
1;
1, 1;
2, 0, 1;
3, 1, 0, 1;
6, 0, 0, 0, 1;
7, 2, 1, 0, 0, 1;
14, 0, 0, 0, 0, 0, 1;
17, 3, 0, 1, 0, 0, 0, 1;
27, 0, 2, 0, 0, 0, 0, 0, 1;
34, 6, 0, 0, 1, 0, 0, 0, 0, 1;
55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
63, 7, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1;
100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
119, 14, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1;
167, 0, 6, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
209, 17, 0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1;
296, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
...
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1, x,
b(n, i-1)+(p-> add(coeff(p, x, t)*x^igcd(t, i),
t=0..degree(p)))(add(b(n-i*j, i-1), j=1..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
seq(T(n), n=1..17); # Alois P. Heinz, Mar 29 2015
MATHEMATICA
b[n_, i_] := b[n, i] = If[n==0, 1, If[i==1, x, b[n, i-1] + Function[{p}, Sum[Coefficient[p, x, t]*x^GCD[t, i], {t, 0, Exponent[p, x]}]][Sum[b[n - i*j, i-1], {j, 1, n/i}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, n]]; Table[T[n], {n, 1, 17}] // Flatten (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Nov 28 2009
EXTENSIONS
Corrected and extended by Alois P. Heinz, Jun 06 2013
STATUS
approved