OFFSET
0,2
COMMENTS
Conjecture 1.1 of Osburn and Sahu is if p is a prime and JacobiSymbol(p, 23) = 1 and n>0 then a(n * p) == a(n) (mod p). - Michael Somos, Sep 21 2013
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Robert Osburn, Brundaban Sahu, Congruences via modular forms, arXiv:0912.0173 [math.NT], (Sep 02 2010).
R. Osburn and B. Sahu, Congruences via modular forms, Proc. Amer. Math. Soc. 139 (2011), 2375-2381.
FORMULA
n^2 * a(n) = (14*n^2 - 21*n + 9) * a(n-1) + (-57*n^2 + 171*n - 136) * a(n-2) + (106*n^2 - 477*n + 551) * a(n-3) + (-90*n^2 + 540*n - 816) * a(n-4) + (16*n^2 - 120*n + 224) * a(n-5) + (19*n^2 - 171*n + 380) * a(n-6). - Michael Somos, Sep 21 2013
G.f. A(x) satisfies f(q) = A(g(q)) where f is the g.f. for A028959 and g(q) = eta(q) * eta(q^23) / f(q). - Michael Somos, Sep 21 2013
EXAMPLE
G.f. = 1 + 2*x + 6*x^2 + 26*x^3 + 142*x^4 + 876*x^5 + 5790*x^6 + 40020*x^7 + ...
MAPLE
p := (1+224*x -864*x^2 -544*x^3 +9664*x^4 -26112*x^5 +36288*x^6 -27648*x^7 +9216*x^8) ;
s := (1-14*x+57*x^2-106*x^3+90*x^4-16*x^5-19*x^6)^(1/2) ;
A := (5*(53-400*x+944*x^2-912*x^3+288*x^4)-24*(11-16*x)*s)/p ;
f := 4*x*(1-45*x+865*x^2-9270*x^3+60648*x^4 -249463*x^5+640904*x^6 -987056*x^7 +821224*x^8-249920*x^9 -71232*x^10+20610*x^11 -(1-21*x +148*x^2 -380*x^3+212*x^4)*(1-17*x+90*x^2-142*x^3 -14*x^4)*s)*(6*A)^3/23^6;
ogf := A^(1/4) * hypergeom([1/12, 5/12], [1], f);
series(ogf, x=0, 101); # Mark van Hoeij, Apr 12 2014
MATHEMATICA
a[ n_] := If[ n < 0, 0, With[ {f = Series[ EllipticTheta[ 3, 0, x] EllipticTheta[ 3, 0, x^23] + EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^23], {x, 0, n}], g = x QPochhammer[ x] QPochhammer[ x^23]}, SeriesCoefficient[ ComposeSeries[ f, InverseSeries[ g/f ], {x, 0, n}]]]; (* Michael Somos, Sep 21 2013 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Joerg Arndt, Apr 09 2013
STATUS
approved