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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. 9
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified March 30 19:49 EDT 2020. Contains 333127 sequences. (Running on oeis4.)