A096562 Coefficients of replicable function number "25a" with a(0) = -1.

1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, -1, 0, 0, 1, 0, 1, 0, 0, -1, 0, -1, 0, 0, -2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, -2, 0, 0, 2, 0, 3, 0, 0, -1, 0, -2, 0, 0, -3, 0, 0, 0, 0, -1, 0, 2, 0, 0, 3, 0, -4, 0, 0, 3, 0, 4, 0, 0, -2, 0, -3, 0, 0, -5, 0, 1, 0, 0, -1, 0, 3, 0, 0, 6, 0, -6, 0, 0

Offset

-1,41

References

G. E. Andrews and B. C. Berndt, Ramanujan's Lost Notebook, Part I, Springer, 2005, see p. 11, Equation (1.1.10)

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

T. Horie and N. Kanou, Certain modular functions similar to the Dedekind eta function, Abh. Math. Sem. Univ. Hamburg 72 (2002), 89-117. MR1941549 (2003j:11043)

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, see p. 238, Equation (20.2)

Links

Seiichi Manyama, Table of n, a(n) for n = -1..10000

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of eta(q) / eta(q^25) = (1/q) * f(-q) / f(-q^25) in powers of q where f() is a Ramanujan theta function.

Euler transform of period 25 sequence [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 0, ...].

G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = (u^2 - v) * (u - v^2) - 2*u*v * (u + v + 2).

G.f. A(q) satisfies 0 = f(A(q), A(q^2), A(q^4)) where f(u, v, w) = u^2 + u*w + w^2 - v*(2*(u + w) + 5) - v^2*(u + w + 2).

G.f.: x^-1 * Product_{k>0} (1 - x^k) / (1 - x^(25*k)).

Convolution inverse of A092885. a(n) = A096563(n) unless n=0.

Expansion of 1/R(q) - 1 - R(q) in powers of q where R() is the g.f. of A007325 the Rogers-Ramanujan continued fraction. - Michael Somos, May 09 2016

a(-1) = 1, a(n) = -(1/(n+1))*Sum_{k=1..n+1} A227131(k)*a(n-k) for n > -1. - Seiichi Manyama, Mar 29 2017

Example

G.f. = 1/q - 1 - q + q^4 + q^6 - q^11 - q^14 + q^21 + q^24 - q^26 + q^29 + ...

Mathematica

a[ n_] := With[ {m = n + 1}, SeriesCoefficient[ Product[ 1 - q^k, {k, m}] / Product[ 1 - q^k, {k, 25, m, 25}], {q, 0, m}]];
a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q] / QPochhammer[ q^25]), {q, 0, n}]; (* Michael Somos, Jul 05 2014 *)

Prog

(PARI) {a(n) = my(A, m); if( n<-1, 0, m=5; A = x + O(x^6); while( m < n + 2, m*=5; A = x * subst((A * (1 - 2*A + 4*A^2 - 3*A^3 + A^4) / (1 + 3*A+ 4*A^2 + 2*A^3 + A^4) / x)^(1/5), x, x^5)); polcoeff( 1/A - A - 1, n))};
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) / eta(x^25 + A), n))};

Crossrefs

Cf. A007325, A010815, A092885, A096563.

Sequence in context: A324179 A372357 A216577 * A096563 A216512 A078359

Adjacent sequences: A096559 A096560 A096561 * A096563 A096564 A096565

Keyword

sign

Author

Michael Somos, Jul 02 2004

Status

approved