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A113261
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Expansion of (9 phi(q) phi(q^3)^5 - phi(q)^5 phi(q^3))/8 in powers of q where phi(q) is a Ramanujan theta function.
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1
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1, 1, -5, 1, 11, -24, -5, 50, -53, 1, 120, -120, 11, 170, -250, -24, 203, -288, -5, 362, -264, 50, 600, -528, -53, 601, -850, 1, 550, -840, 120, 962, -821, -120, 1440, -1200, 11, 1370, -1810, 170, 1272, -1680, -250, 1850, -1320, -24, 2640, -2208, 203, 2451, -3005, -288, 1870, -2808
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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REFERENCES
| B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 226 Entry 4(ii).
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Euler transform of period 12 sequence [1, -6, 6, -5, 1, -12, 1, -5, 6, -6, 1, -6, ...].
Expansion of et(q^2)^7 eta(q^6)^11/(eta(q)eta(q^3)^5 eta(q^4)eta(q^12)^5) in powers of q.
a(n) is multiplicative and a(3^e) = 1, a(2^e) = ((-4)^(e+1)-1)/(-4-1)-2, a(p^e) = ((-p^2)^(e+1)-1)/(-p-1) if p == 2 (mod 3), a(p^e) = ((p^2)(e+1)-1)/(p-1) if p == 1 (mod 3).
G.f.: 1 + Sum_{k>0} k^2 x^k/(1-(-x)^k) kronecker(-3, k).
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PROG
| (PARI) a(n)=if(n<1, n==0, sumdiv(n, d, d^2*kronecker(-3, d)*(-1)^(n-d)))
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^7*eta(x^6+A)^11/(eta(x+A)*eta(x^3+A)^5*eta(x^4+A)*eta(x^12+A)^5), n))}
(PARI) {a(n)=local(A, p, e, t); if(n<1, n==0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==3, 1, t=p^2*(-1)^(p%3==2); (t^(e+1)-1)/(t-1)-2*(p==2)))))}
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CROSSREFS
| Sequence in context: A159074 A147414 A117637 * A132000 A132001 A063004
Adjacent sequences: A113258 A113259 A113260 * A113262 A113263 A113264
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KEYWORD
| sign,mult
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AUTHOR
| Michael Somos, Oct 21 2005
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