A224529 - OEIS

%I #36 May 30 2019 00:41:13

%S 1,2,6,26,142,876,5790,40020,285582,2087612,15551620,117629724,

%T 900964558,6973745924,54464010540,428645647572,3396238954446,

%U 27067890450300,216857021933172,1745460025192140,14107695302434356,114455036696796168,931738743735004596

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

%C 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

%H Alois P. Heinz, <a href="/A224529/b224529.txt">Table of n, a(n) for n = 0..1000</a>

%H Robert Osburn, Brundaban Sahu, <a href="http://arxiv.org/abs/0912.0173">Congruences via modular forms</a>, arXiv:0912.0173 [math.NT], (Sep 02 2010).

%H R. Osburn and B. Sahu, <a href="https://doi.org/10.1090/S0002-9939-2010-10771-2">Congruences via modular forms</a>, Proc. Amer. Math. Soc. 139 (2011), 2375-2381.

%F 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

%F 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

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

%p 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) ;

%p s := (1-14*x+57*x^2-106*x^3+90*x^4-16*x^5-19*x^6)^(1/2) ;

%p A := (5*(53-400*x+944*x^2-912*x^3+288*x^4)-24*(11-16*x)*s)/p ;

%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;

%p ogf := A^(1/4) * hypergeom([1/12, 5/12],[1], f);

%p series(ogf, x=0, 101);  # _Mark van Hoeij_, Apr 12 2014

%t 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 *)

%Y Cf. A224530 (sequence F_n).

%Y Cf. A028959, A030199.

%K nonn

%O 0,2

%A _Joerg Arndt_, Apr 09 2013