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URL: https://oeis.org/A277076

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A277076
Expansion of f(-x)^8 * Q(x) in powers of x where f() is a Ramanujan theta function and Q() is a Ramanujan Lambert series.
1
1, 232, 260, -5760, 6890, 7744, 33176, -115200, 14035, 60320, 1508, 449280, -380770, -599040, 7640, 599040, -755943, 1598480, 1843200, -2620800, -988858, -2995712, 3857360, -1497600, -2004730, 7696832, 2699684, 1670400, -7188480, -11980800, 1791400, 10736640
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Ramanujan Lambert series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).
REFERENCES
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 329, 2nd equation.
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of f(-x)^8 * (f(-x)^24 + 256 * x * f(-x^2)^24) / (f(-x) * f(-x^2))^8 in powers of x.
a(n) = b(3*n+1) where b() is multiplicative with b(p^e) = 0^e if p=3 and b(p^e) = b(p)*b(p^(e-1)) - p^7*b(p^(e-2)) otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 6561 (t/i)^8 f(t) where q = exp(2 Pi i t).
Convolution of A000731 and A004009.
EXAMPLE
G.f. = 1 + 232*x + 260*x^2 - 5760*x^3 + 6890*x^4 + 7744*x^5 + 33176*x^6 - 115200*x^7 + 14035*x^8 + ...
G.f. = q + 232*q^4 + 260*q^7 - 5760*q^10 + 6890*q^13 + 7744*q^16 + 33176*q^19 - 115200*q^22 + ...
MATHEMATICA
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ QPochhammer[ x]^8 (1 + 240 Sum[ DivisorSigma[ 3, k] x^k, {k, n}]), {x, 0, n}]];
a[ n_] := SeriesCoefficient[ With[ {A1 = QPochhammer[ x]^8, A2 = QPochhammer[ x^2]^8}, A1 (A1^3 + 256 x A2^3) / (A1 A2)], {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^8 * sum(k=1, n, 240 * sigma(k, 3) * x^k, 1 + A), n))};
(PARI) {a(n) = my(A, A1, A2); if( n<0, 0, A = x * O(x^n); A1 = eta(x + A)^8; A2 = eta(x^2 + A)^8; polcoeff( A1 * (A1^3 + 256 * x * A2^3) / (A1 * A2), n))};
(Magma) A := Basis( CuspForms( Gamma0(9), 8), 95); A[1] + 232*A[4];
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Sep 27 2016
STATUS
approved