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

0,2

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

Cf. A224530 (sequence F_n).

Cf. A028959, A030199.

Sequence in context: A009575 A263687 A180891 * A171151 A177381 A289312

Adjacent sequences: A224526 A224527 A224528 * A224530 A224531 A224532

Keyword

nonn

Author

Joerg Arndt, Apr 09 2013

Status

approved