A078905 The q expansion of Lambda^5, a Hauptmodul for Gamma_1(5).

1, -5, 15, -30, 40, -26, -30, 125, -220, 245, -124, -180, 615, -1010, 1085, -550, -705, 2415, -3850, 3980, -1926, -2460, 8090, -12550, 12715, -6074, -7500, 24360, -37150, 36930, -17251, -21155, 67380, -101210, 99295, -45924, -55305, 174500, -259140, 251275, -114750

Offset

1,2

Comments

Denoted by r^5(tau) by Duke (2005). - Michael Somos, Jul 09 2014

References

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

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

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1001 from T. D. Noe)

W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162. See page 150.

Formula

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

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.

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

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

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

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

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

Example

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

Mathematica

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 *)

Prog

(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))};
(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))};
(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))};

Crossrefs

Cf. A007325, A109064.

Sequence in context: A188350 A268222 A285630 * A059160 A319930 A357690

Adjacent sequences: A078902 A078903 A078904 * A078906 A078907 A078908

Keyword

sign,easy,nice

Author

Michael Somos, Dec 12 2002

Status

approved