A132967 - OEIS

%I #15 Mar 12 2021 22:24:44

%S 1,0,1,-1,1,-2,2,-2,3,-4,4,-5,6,-6,9,-11,10,-14,16,-17,22,-24,26,-32,

%T 37,-40,47,-54,58,-70,80,-84,100,-112,122,-143,158,-172,198,-222,242,

%U -274,306,-332,379,-422,454,-515,569,-620,698,-766,834,-932,1028,-1118

%N Expansion of q * chi(-q^3) * chi(-q^5) / ( chi(-q^2) * chi(-q^30) ) in powers of q where chi() is a Ramanujan theta function.

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

%H Seiichi Manyama, <a href="/A132967/b132967.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/retaprod.html">A Remarkable eta-product Identity</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of eta(q^3) * eta(q^4) * eta(q^5) * eta(q^60) / (eta(q^2) * eta(q^6) * eta(q^10) * eta(q^30)) in powers of q.

%F Euler transform of period 60 sequence [0, 1, -1, 0, -1, 1, 0, 0, -1, 1, 0, 0, 0, 1, -2, 0, 0, 1, 0, 0, -1, 1, 0, 0, -1, 1, -1, 0, 0, 2, 0, 0, -1, 1, -1, 0, 0, 1, -1, 0, 0, 1, 0, 0, -2, 1, 0, 0, 0, 1, -1, 0, 0, 1, -1, 0, -1, 1, 0, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132968.

%F G.f.: x * Product_{k>0} (1 + x^(2*k)) * (1 + x^(30*k)) / ( (1 + x^(3*k)) * (1 + x^(5*k)) ).

%F a(n) = - A132968(n) unless n=0.

%e G.f. = q + q^3 - q^4 + q^5 - 2*q^6 + 2*q^7 - 2*q^8 + 3*q^9 - 4*q^10 + 4*q^11 - ...

%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x*O(x^n); polcoeff( eta(x^3 + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^60 + A) / (eta(x^2 + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^30 + A)), n))};

%Y Cf. A132968.

%K sign

%O 1,6

%A _Michael Somos_, Sep 02 2007