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%I #26 Jan 11 2024 16:06:33
%S 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,
%T 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,
%U 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
%N Triangle read by rows, A054525 * A168021.
%C Row sums = A000041 starting (1, 2, 3, 5, 7, 11, 15, ...).
%C T(n,k) is the number of partitions of n into parts with GCD = k. - _Alois P. Heinz_, Jun 06 2013
%H Alois P. Heinz, <a href="/A168532/b168532.txt">Rows n = 1..141, flattened</a>
%F Mobius transform of triangle A168021 = an infinite lower triangular matrix with aerated variants of A000837 in each column; where A000837 = the Mobius transform of the partition numbers, A000041.
%e First few rows of the triangle:
%e 1;
%e 1, 1;
%e 2, 0, 1;
%e 3, 1, 0, 1;
%e 6, 0, 0, 0, 1;
%e 7, 2, 1, 0, 0, 1;
%e 14, 0, 0, 0, 0, 0, 1;
%e 17, 3, 0, 1, 0, 0, 0, 1;
%e 27, 0, 2, 0, 0, 0, 0, 0, 1;
%e 34, 6, 0, 0, 1, 0, 0, 0, 0, 1;
%e 55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 63, 7, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1;
%e 100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 119, 14, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1;
%e 167, 0, 6, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 209, 17, 0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1;
%e 296, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e ...
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1, x,
%p b(n, i-1)+(p-> add(coeff(p, x, t)*x^igcd(t, i),
%p t=0..degree(p)))(add(b(n-i*j, i-1), j=1..n/i))))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
%p seq(T(n), n=1..17); # _Alois P. Heinz_, Mar 29 2015
%t 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_ *)
%Y Cf. A168021, A000837.
%Y Cf. A256067 (the same for LCM).
%K nonn,tabl
%O 1,4
%A _Gary W. Adamson_, Nov 28 2009
%E Corrected and extended by _Alois P. Heinz_, Jun 06 2013
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