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A094362
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McKay-Thompson series of class 39C for the Monster group with a(0) = 1.
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3
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1, 1, 2, 2, 4, 5, 7, 9, 13, 16, 22, 27, 36, 43, 56, 68, 87, 104, 130, 156, 193, 230, 281, 333, 404, 477, 572, 673, 802, 940, 1113, 1299, 1531, 1780, 2085, 2418, 2820, 3259, 3784, 4362, 5047, 5799, 6685, 7662, 8806, 10066, 11532, 13152, 15026, 17098, 19482
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OFFSET
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-1,3
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LINKS
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FORMULA
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Expansion of (eta(q^3) * eta(q^13)) / (eta(q) * eta(q^39)) in powers of q.
Euler transform of period 39 sequence [ 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) = f(1/A(x), 1/A(x^2)) where f(u, v) = u^3 + v^3 + 2*u*v*(u + v) - u^2*v^2 - u*v.
G.f.: x^-1 * Product_{k>0} (1 - x^(3*k)) * (1 - x^(13*k)) / ((1 - x^k) * (1 - x^(39*k))).
a(n) ~ exp(4*Pi*sqrt(n/39)) / (sqrt(2) * 39^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 06 2015
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EXAMPLE
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1/q + 1 + 2*q + 2*q^2 + 4*q^3 + 5*q^4 + 7*q^5 + 9*q^6 + 13*q^7 + 16*q^8 + 22*q^9 + ...
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MATHEMATICA
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QP = QPochhammer; s = QP[q^3]*(QP[q^13]/(QP[q]*QP[q^39])) + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, adapted from PARI *)
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PROG
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(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^13 + A) / (eta(x + A) * eta(x^39 + A)), n))}
(PARI) {a(n) = local(A, u, v); if( n<-1, 0, A = 1/x; for( k=0, n, u = A + x * O(x^k); v = subst(u, x, x^2); A += x^k * polcoeff( u^3 + v^3 + 2*u*v*(u + v) - u^2*v^2 - u*v, k-5) / 2); polcoeff(A, n))}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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