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, 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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 329, 2nd equation.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

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

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

Cf. A000731, A000739, A004009.

Sequence in context: A280538 A050423 A126818 * A250645 A179246 A064715

Adjacent sequences: A277073 A277074 A277075 * A277077 A277078 A277079

Keyword

sign

Author

Michael Somos, Sep 27 2016

Status

approved