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%I #12 Sep 08 2022 08:46:17
%S 1,232,260,-5760,6890,7744,33176,-115200,14035,60320,1508,449280,
%T -380770,-599040,7640,599040,-755943,1598480,1843200,-2620800,-988858,
%U -2995712,3857360,-1497600,-2004730,7696832,2699684,1670400,-7188480,-11980800,1791400,10736640
%N 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.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Ramanujan Lambert series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).
%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 329, 2nd equation.
%H G. C. Greubel, <a href="/A277076/b277076.txt">Table of n, a(n) for n = 0..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of f(-x)^8 * (f(-x)^24 + 256 * x * f(-x^2)^24) / (f(-x) * f(-x^2))^8 in powers of x.
%F 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.
%F 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).
%F Convolution of A000731 and A004009.
%e 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 + ...
%e G.f. = q + 232*q^4 + 260*q^7 - 5760*q^10 + 6890*q^13 + 7744*q^16 + 33176*q^19 - 115200*q^22 + ...
%t a[ n_] := If[ n < 0, 0, SeriesCoefficient[ QPochhammer[ x]^8 (1 + 240 Sum[ DivisorSigma[ 3, k] x^k, {k, n}]), {x, 0, n}]];
%t 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}];
%o (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))};
%o (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))};
%o (Magma) A := Basis( CuspForms( Gamma0(9), 8), 95); A[1] + 232*A[4];
%Y Cf. A000731, A000739, A004009.
%K sign
%O 0,2
%A _Michael Somos_, Sep 27 2016
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