A074586 - OEIS

%I #23 Dec 12 2015 13:11:40

%S 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,

%T 19,67,108,98,53,16,2,1,22,90,177,206,151,69,18,2,1,26,123,282,393,

%U 359,220,87,20,2,1,28,149,405,675,752,579,307,107,22,2,1,34,203,594,1109

%N 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.

%H T. D. Noe, <a href="/A074586/b074586.txt">Rows n=0..50 of triangle, flattened</a>

%F 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).

%F 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.

%e The first few Moebius polynomials are as follows:

%e M(1,x) = 1;

%e M(2,x) = 1 + 2*x;

%e M(3,x) = 1 + 4*x + 2*x^2;

%e M(4,x) = 1 + 7*x + 8*x^2 + 2*x^3;

%e M(5,x) = 1 + 9*x + 15*x^2 + 10*x^3 + 2*x^4;

%e M(6,x) = 1 + 13*x + 30*x^2 + 27*x^3 + 12*x^4 + 2*x^5;

%e M(7,x) = 1 + 15*x + 43*x^2 + 57*x^3 + 39*x^4 + 14*x^5 + 2*x^6; ...

%e ILLUSTRATION OF GENERATING METHOD:

%e M(1,x) = 1;

%e M(2,x) = 1 + 2*x*M(1,x) = 1 + 2*x;

%e 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;

%e 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;

%e 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; ...

%e This triangle of coefficients begins:

%e   1

%e   1  2

%e   1  4   2

%e   1  7   8   2

%e   1  9  15  10    2

%e   1 13  30  27   12    2

%e   1 15  43  57   39   14    2

%e   1 19  67 108   98   53   16   2

%e   1 22  90 177  206  151   69  18   2

%e   1 26 123 282  393  359  220  87  20   2

%e   1 28 149 405  675  752  579 307 107  22  2

%e   1 34 203 594 1109 1439 1333 886 414 129 24 2 ...

%t 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 *)

%o (PARI) {T(n,k)=if(k==1,1,sum(m=1,n-1,floor(n/m)*T(m,k-1)))}

%o for(n=1,12, for(k=1,n, print1(T(n,k),", "));print(""))

%Y Cf. A074587.

%K easy,nice,nonn,tabl

%O 1,3

%A _Paul D. Hanna_, Aug 25 2002