|
%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
|
