OFFSET
1,3
LINKS
T. D. Noe, Rows n=0..50 of triangle, flattened
FORMULA
The n-th row consists of the coefficients of M(n, x) as a polynomial in x, where M(n, x) = 1 + [n/1]*x*M(1, x) + [n/2]*x*M(2, x) + [n/3]*x*M(3, x) +... + [n/(n-1)]*x*M(n-1, x) for n>1, with M(1, x) = 1, where [x] = floor(x).
T(n, k) = Sum_{m=1..n-1} [n/m]*T(m, k-1) for n>=k>1, with T(n, 1)=1 for n>=1.
EXAMPLE
The first few Moebius polynomials are as follows:
M(1,x) = 1;
M(2,x) = 1 + 2*x;
M(3,x) = 1 + 4*x + 2*x^2;
M(4,x) = 1 + 7*x + 8*x^2 + 2*x^3;
M(5,x) = 1 + 9*x + 15*x^2 + 10*x^3 + 2*x^4;
M(6,x) = 1 + 13*x + 30*x^2 + 27*x^3 + 12*x^4 + 2*x^5;
M(7,x) = 1 + 15*x + 43*x^2 + 57*x^3 + 39*x^4 + 14*x^5 + 2*x^6; ...
ILLUSTRATION OF GENERATING METHOD:
M(1,x) = 1;
M(2,x) = 1 + 2*x*M(1,x) = 1 + 2*x;
M(3,x) = 1 + 3*x*M(1,x) + [3/2]*x*M(2,x) = 1 + 3*x + x*(1+2*x) = 1 + 4*x + 2*x^2;
M(4,x) = 1 + 4*x*M(1,x) + [4/2]*x*M(2,x) + [4/3]*x*M(3,x) = 1 + 4*x + 2*x*(1 + 2*x) + 1*x*(1 + 4*x + 2*x^2) = 1 + 7*x + 8*x^2 + 2*x^3;
M(5,x) = 1 + 5*x*M(1,x) + [5/2]*x*M(2,x) + [5/3]*x*M(3,x) + [5/4]*x*M(4,x) = 1 + 5*x + 2*x*(1 + 2*x) + 1*x*(1 + 4*x + 2*x^2) + 1*x*(1 + 7*x + 8*x^2 + 2*x^3) = 1 + 9*x + 15*x^2 + 10*x^3 + 2*x^4; ...
This triangle of coefficients begins:
1
1 2
1 4 2
1 7 8 2
1 9 15 10 2
1 13 30 27 12 2
1 15 43 57 39 14 2
1 19 67 108 98 53 16 2
1 22 90 177 206 151 69 18 2
1 26 123 282 393 359 220 87 20 2
1 28 149 405 675 752 579 307 107 22 2
1 34 203 594 1109 1439 1333 886 414 129 24 2 ...
MATHEMATICA
t[n_, 1] = 1; t[n_, k_] := t[n, k] = Sum[ Floor[n/m]*t[m, k-1], {m, 1, n-1}]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 03 2012, after PARI *)
PROG
(PARI) {T(n, k)=if(k==1, 1, sum(m=1, n-1, floor(n/m)*T(m, k-1)))}
for(n=1, 12, for(k=1, n, print1(T(n, k), ", ")); print(""))
CROSSREFS
KEYWORD
AUTHOR
Paul D. Hanna, Aug 25 2002
STATUS
approved