A164269 - OEIS

%I #20 Mar 12 2021 22:24:46

%S 1,2,0,-7,-12,0,32,50,0,-114,-168,0,350,496,0,-967,-1332,0,2468,3324,

%T 0,-5916,-7824,0,13471,17548,0,-29384,-37788,0,61784,78578,0,-125838,

%U -158496,0,249230,311224,0,-481506,-596676,0,909788,1119624,0,-1684824,-2060448,0

%N Expansion of q * f(q^9)^3 * phi(q) / (f(q^3) * phi(q^3)^3) in powers of q where f(), phi() are Ramanujan theta functions.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A164269/b164269.txt">Table of n, a(n) for n = 1..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 Euler transform of period 36 sequence [2, -3, -5, -1, 2, 8, 2, -1, -2, -3, 2, 3, 2, -3, -5, -1, 2, 2, 2, -1, -5, -3, 2, 3, 2, -3, -2, -1, 2, 8, 2, -1, -5, -3, 2, 0, ...].

%F a(3*n) = 0. a(3*n + 1) = A164270(n). a(3*n + 2) = 2 * A164271(n).

%F Convolution inverse of A164268.

%F Expansion of x * phi(x) * chi(x^3) * f(x^9)^3 / phi(x^3)^4 = x * phi(x) * f(x^9)^3 / (chi(x^3)^3 * f(x^3)^4) in powers of x where phi(), chi(), f() are Ramanujan theta functions. - _Michael Somos_, Sep 20 2017

%e G.f. = q + 2*q^2 - 7*q^4 - 12*q^5 + 32*q^7 + 50*q^8 - 114*q^10 - 168*q^11 + ...

%t f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A164269[n_] := SeriesCoefficient[q*(f[q^9, -q^18]^3*f[q, q])/(( f[q^3, -q^6])*f[q^3, q^3]^3), {q, 0, n}]; Rest[Table[A164269[n], {n,0,50}]] (* _G. C. Greubel_, Sep 16 2017 *)

%t a[ n_] := SeriesCoefficient[ x QPochhammer[ -x^9]^3 EllipticTheta[ 3, 0, x] / (QPochhammer[ -x^3] EllipticTheta[ 3, 0, x^3]^3), {x, 0, n}]; (* _Michael Somos_, Sep 17 2017 *)

%t a[ n_] := SeriesCoefficient[ x QPochhammer[ -x^3, x^6] QPochhammer[ x^9]^3 EllipticTheta[ 3, 0, x] / EllipticTheta[ 3, 0, x^3]^4, {x, 0, n}]; /(* _Michael Somos_, Sep 20 2017 *)

%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^7 * eta(x^12 + A)^7 * eta(x^18 + A)^9 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^18 * eta(x^9 + A)^3 * eta(x^36 + A)^3), n))};

%Y Cf. A164268, A164270, A164271.

%K sign

%O 1,2

%A _Michael Somos_, Aug 11 2009