A096562 - OEIS

%I #28 Oct 06 2019 18:09:36

%S 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,

%T 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,

%U 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

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

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

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

%D 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)

%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, see p. 238, Equation (20.2)

%H Seiichi Manyama, <a href="/A096562/b096562.txt">Table of n, a(n) for n = -1..10000</a>

%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).

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

%F 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.

%F 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, ...].

%F 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).

%F 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).

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

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

%F 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

%F 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

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

%t a[ n_] := With[ {m = n + 1}, SeriesCoefficient[ Product[ 1 - q^k, {k, m}] / Product[ 1 - q^k, {k, 25, m, 25}], {q, 0, m}]];

%t a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q] / QPochhammer[ q^25]), {q, 0, n}]; (* _Michael Somos_, Jul 05 2014 *)

%o (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))};

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) / eta(x^25 + A), n))};

%Y Cf. A007325, A010815, A092885, A096563.

%K sign

%O -1,41

%A _Michael Somos_, Jul 02 2004