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A145725
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McKay-Thompson series of class 60C for the Monster group with a(0) = 1.
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2
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1, 1, 1, 1, 2, 2, 2, 3, 5, 5, 5, 7, 9, 10, 11, 14, 18, 20, 22, 27, 32, 36, 40, 48, 57, 63, 70, 82, 95, 106, 119, 137, 158, 175, 195, 222, 252, 280, 311, 352, 397, 439, 486, 546, 611, 676, 747, 834, 929, 1024, 1128, 1253, 1389, 1528, 1679, 1857, 2052, 2250, 2467, 2718
(list; graph; refs; listen; history; internal format)
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OFFSET
| -1,5
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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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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of q^(-1) * psi(-q^3) * psi(-q^5) / (psi(-q) * psi(-q^15)) in powers of q where psi() is a Ramanujan theta function.
Expansion of eta(q^2) * eta(q^3) * eta(q^5) * eta(q^12) * eta(q^20) * eta(q^30) / (eta(q) * eta(q^4) * eta(q^6) * eta(q^10) * eta(q^15) * eta(q^60)) in powers of q.
Euler transform of period 60 sequence [ 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 - u - v - u*v) * (u^3 + u^2*v + u*v^2 +v^3) + u*v * (1 + u^2) * (1 + v^2) + 2*u^2*v^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = f(t) where q = exp(2 pi i t).
G.f.: 1 / ( x * Product_{k>0} P(15, x^k) * P(60, x^k) ) where P(n, x) is the n-th cyclotomic polynomial.
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EXAMPLE
| 1/q + 1 + q + q^2 + 2*q^3 + 2*q^4 + 2*q^5 + 3*q^6 + 5*q^7 + 5*q^8 + ...
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PROG
| (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^5 + A) * eta(x^12 + A) * eta(x^20 + A) * eta(x^30 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^15 + A) * eta(x^60 + A)), n))}
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CROSSREFS
| A058727(n) = a(n) unless n=0. A135213(n) = -(-1)^n * a(n).
Sequence in context: A029073 A058618 A058727 * A135213 A035658 A077018
Adjacent sequences: A145722 A145723 A145724 * A145726 A145727 A145728
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Oct 18 2008
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