1,2

Setting x=1 gives us phi(n) (A000010). Setting x=0 gives A092674.

Let t(n)=binomial(n+1,2); s[1]=x; for i >= 2, s[i]=t(i)-sum_{j=1..i-1} s[j]*floor(i/j); a(n) = coefficient of x in s[n]. - Jon Perry

a(n) is multiplicative with a(2)= -2, a(4)= 1, a(2^e)= 0 if e>2. a(p)= -1, a(p^e)= 0 if e>1, p>2. - Michael Somos Mar 26 2007

a(8n)= 0 . a(2n+1)= moebius(2n+1) . a(2n)= moebius(2n)- moebius(n) . - Michael Somos Mar 26 2007

|a(n)|<=2 .

1/(1+x)= Product_{k>0} f(-x^k)^a(k) where f() is a Ramanujan theta function. - Michael Somos Mar 26 2007

The first few s[n] are:

x, -2*x + 3, -x + 3, x + 1, -x + 5, 2*x, -x + 7, 4, 6, 2*x + 2, -x + 11, -x + 5, -x + 13, 2*x + 4, x + 7, 8, -x + 17, 6, -x + 19, -x + 9, x + 11, 2*x + 8, -x + 23, 8, 20, 2*x + 10, 18, -x + 13, -x + 29, -2*x + 10, -x + 31, 16, x + 19.

(PARI) s=vector(2000); t(n)=binomial(n+1, 2); s[1]=x; for(i=2, 2000, s[i]=t(i)-sum(j=1, i-1, s[j]*floor(i/j))); for(i=1, 2000, print1(", "polcoeff(s[i], 1)))

(PARI) {a(n)= if(n<1, 0, moebius(n)- if(n%2, 0, moebius(n/2)))} /* Michael Somos Mar 26 2007 */

(PARI) {a(n)= local(A, B, m); if(n<1, 0, A= x*O(x^n); B= 1+x+A; for(k=1, n, B*= eta(x^k+A)^ (m= polcoeff(B, k))); m)} /* Michael Somos Mar 26 2007 */

Cf. A092674.

Sequence in context: A143262 A035180 A163819 this_sequence A112400 A116861 A105242

Adjacent sequences: A092670 A092671 A092672 this_sequence A092674 A092675 A092676

sign,mult

Jon Perry (perry(AT)globalnet.co.uk), Mar 02 2004