A224529 Sequence f_n from a paper by Robert Osburn and Brundaban Sahu.

1, 2, 6, 26, 142, 876, 5790, 40020, 285582, 2087612, 15551620, 117629724, 900964558, 6973745924, 54464010540, 428645647572, 3396238954446, 27067890450300, 216857021933172, 1745460025192140, 14107695302434356, 114455036696796168, 931738743735004596

Offset: 0

Author

Joerg Arndt, Apr 09 2013

Keywords

• nonn

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

Examples

			G.f. = 1 + 2*x + 6*x^2 + 26*x^3 + 142*x^4 + 876*x^5 + 5790*x^6 + 40020*x^7 + ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Robert Osburn and 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.

Crossrefs

Cf. A224530 (sequence F_n).

Cf. A028959, A030199.

Programs

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 *)

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