A074586 Triangle of Moebius polynomial coefficients, read by rows, the n-th row forming the polynomial M(n,x) such that M(n,-1) = mu(n), the Moebius function of n.

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

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

Cf. A074587.

Sequence in context: A133938 A239829 A210034 * A277812 A134586 A135287

Adjacent sequences: A074583 A074584 A074585 * A074587 A074588 A074589

Keyword

easy,nice,nonn,tabl

Author

Paul D. Hanna, Aug 25 2002

Status

approved