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Search: id:A035175
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| A035175 |
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Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -15. |
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+0
4
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| 1, 2, 1, 3, 1, 2, 0, 4, 1, 2, 0, 3, 0, 0, 1, 5, 2, 2, 2, 3, 0, 0, 2, 4, 1, 0, 1, 0, 0, 2, 2, 6, 0, 4, 0, 3, 0, 4, 0, 4, 0, 0, 0, 0, 1, 4, 2, 5, 1, 2, 2, 0, 2, 2, 0, 0, 2, 0, 0, 3, 2, 4, 0, 7, 0, 0, 0, 6, 2, 0, 0, 4, 0, 0, 1, 6, 0, 0, 2, 5, 1, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 6, 2, 4, 2, 6, 0, 2, 0, 3, 0, 4, 0, 0, 0
(list; graph; listen)
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OFFSET
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1,2
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FORMULA
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Expansion of -1+(eta(q^3)eta(q^5))^2/(eta(q)eta(q^15)) in powers of q. - Michael Somos Aug 25 2006
Euler transform of period 15 sequence [ 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -2, ...]. if a(0)=1. - Michael Somos Aug 25 2006
Moebius transform is period 15 sequence [ 1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, -1, 0, ...]. - Michael Somos Aug 25 2006
Given g.f. A(x), then B(x)=1+A(x) satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=-v^3+4uvw-2uw^2-u^2w.
G.f.: -1+x*Product_{k>0} ((1-x^(3k))(1-x^(5k)))^2/((1-x^k)(1-x^(15k))) . - Michael Somos Aug 25 2006
G.f.: -1+(1/2)(Sum_{n,m} x^(n^2+nm+4m^2) +x^(2n^2+nm+2m^2)). - Michael Somos Aug 25 2006
a(n) is multiplicative with a(3^e) = a(5^e) = 1, a(p^e) = (1+(-1)^e)/2 if p == 7, 11, 13, 14 (mod 15), a(p^e) = e+1 if p == 1, 2, 4, 8 (mod 15). - Michael Somos Aug 25 2006
a(15n+7)=a(15n+11)=a(15n+13)=a(15n+14)=0.
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EXAMPLE
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q + 2*q^2 + q^3 + 3*q^4 + q^5 + 2*q^6 + 4*q^8 + q^9 + 2*q^10 +...
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PROGRAM
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(PARI) direuler(p=2, 101, 1/(1-(kronecker(m, p)*(X-X^2))-X))
(PARI) {a(n)=if(n<1, 0, sumdiv(n, d, kronecker(-15, d)))} /* Michael Somos Aug 25 2006 */
(PARI) {a(n)=local(A, p, e); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==3|p==5, 1, if((p%15)!=2^valuation(p%15, 2), (e+1)%2, (e+1))))))} /* Michael Somos Aug 25 2006 */
(PARI) {a(n)=if(n<1, 0, (qfrep([2, 1; 1, 8], n, 1)+qfrep([4, 1; 1, 4], n, 1))[n])} /* Michael Somos Aug 25 2006 */
(PARI) {a(n)=local(A); if(n<1, 0, A=x*O(x^n); polcoeff( eta(x^3+A)^2*eta(x^5+A)^2/eta(x+A)/eta(x^15+A), n))} /* Michael Somos Aug 25 2006 */
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CROSSREFS
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Sequence in context: A129761 A156248 A123864 this_sequence A106406 A092412 A078734
Adjacent sequences: A035172 A035173 A035174 this_sequence A035176 A035177 A035178
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KEYWORD
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nonn,mult
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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