%I #30 Apr 02 2017 09:42:11

%S 1,-5,15,-30,40,-26,-30,125,-220,245,-124,-180,615,-1010,1085,-550,

%T -705,2415,-3850,3980,-1926,-2460,8090,-12550,12715,-6074,-7500,24360,

%U -37150,36930,-17251,-21155,67380,-101210,99295,-45924,-55305,174500,-259140,251275,-114750

%N The q expansion of Lambda^5, a Hauptmodul for Gamma_1(5).

%C Denoted by r^5(tau) by Duke (2005). - _Michael Somos_, Jul 09 2014

%D A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 24.

%D B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 12, Entry 1(ii).

%H Seiichi Manyama, <a href="/A078905/b078905.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1001 from T. D. Noe)

%H W. Duke, <a href="http://dx.doi.org/10.1090/S0273-0979-05-01047-5">Continued fractions and modular functions</a>, Bull. Amer. Math. Soc. 42 (2005), 137-162. See page 150.

%F G.f.: x * ( Product_{k>0} (1 - x^{5*k - 1}) * (1 - x^{5*k - 4}) / ((1 - x^{5*k - 2}) * (1 - x^{5*k - 3})) )^5

%F G.f.: x * ((Sum_{k in Z} (-1)^k * x^((5*k + 3) * k/2)) / (Sum_{k in Z} (-1)^k * x^((5*k + 1) * k/2)))^5.

%F G.f. A(x) = x * B(x)^5 where B(x) is the g.f. of A007325.

%F Euler transform of period 5 sequence [ -5, 5, 5, -5, 0, ...].

%F G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u,v) = u^2 - v + u*v^3 + u^3*v^2 + 10*u*v * (1 - u + v + u*v). - _Michael Somos_, Mar 09 2004

%F Given g.f. A(q), then q * A'(q) / A(q) = g.f. of A109064. [Duke (2005)] - _Michael Somos_, Jul 09 2014

%F a(1) = 1, a(n) = -(5/(n-1))*Sum_{k=1..n-1} A109091(k)*a(n-k) for n > 1. - _Seiichi Manyama_, Apr 01 2017

%e G.f. = q - 5*q^2 + 15*q^3 - 30*q^4 + 40*q^5 - 26*q^6 - 30*q^7 + 125*q^8 + ...

%t QP = QPochhammer; s = (QP[q, q^5]*(QP[q^4, q^5]/(QP[q^2, q^5]*QP[q^3, q^5]) ))^5 + O[q]^50; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 25 2015, from g.f. of A007325 *)

%o (PARI) {a(n) = local(k); if( n<1, 0, k = (7 + sqrtint(40*n - 32)) \ 10; polcoeff( x * (sum(i=-k, k, (-1)^i * x^((5*i^2 + 3*i) / 2), O(x^n)) / sum(i=-k, k, (-1)^i * x^((5*i^2 + i) / 2), O(x^n)))^5, n))};

%o (PARI) {a(n) = local(A); if( n<1, 0, A=O(x^n); A = (eta(x + A) / eta(x^5 + A))^6 / x; polcoeff( 2 / (11 + A + sqrt(125 + 22*A + A^2)), n))};

%o (PARI) {a(n) = local(A, u, v); if( n<0, 0, A=x; for(k=2, n, u = A + x*O(x^k); v = subst(u, x, x^2); A -= x^k * polcoeff( u^2 - v + u*v^3 + u^3*v^2 + 10*u*v * (1 - u + v + u*v), k+1) / 2); polcoeff(A, n))};

%Y Cf. A007325, A109064.

%K sign,easy,nice

%O 1,2

%A _Michael Somos_, Dec 12 2002