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

%I A224529 #42 Jun 30 2025 04:18:02
%S A224529 1,2,6,26,142,876,5790,40020,285582,2087612,15551620,117629724,
%T A224529 900964558,6973745924,54464010540,428645647572,3396238954446,
%U A224529 27067890450300,216857021933172,1745460025192140,14107695302434356,114455036696796168,931738743735004596
%N A224529 Sequence f_n from a paper by Robert Osburn and Brundaban Sahu.
%C A224529 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 A224529 Alois P. Heinz, <a href="/A224529/b224529.txt">Table of n, a(n) for n = 0..1000</a>
%H A224529 Robert Osburn and Brundaban Sahu, <a href="http://arxiv.org/abs/0912.0173">Congruences via modular forms</a>, arXiv:0912.0173 [math.NT], (Sep 02 2010).
%H A224529 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 A224529 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 A224529 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 A224529 G.f. = 1 + 2*x + 6*x^2 + 26*x^3 + 142*x^4 + 876*x^5 + 5790*x^6 + 40020*x^7 + ...
%p A224529 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 A224529 s := (1-14*x+57*x^2-106*x^3+90*x^4-16*x^5-19*x^6)^(1/2) ;
%p A224529 A := (5*(53-400*x+944*x^2-912*x^3+288*x^4)-24*(11-16*x)*s)/p ;
%p A224529 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 A224529 ogf := A^(1/4) * hypergeom([1/12, 5/12],[1], f);
%p A224529 series(ogf, x=0, 101);  # _Mark van Hoeij_, Apr 12 2014
%t A224529 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 A224529 Cf. A224530 (sequence F_n).
%Y A224529 Cf. A028959, A030199.
%K A224529 nonn
%O A224529 0,2
%A A224529 _Joerg Arndt_, Apr 09 2013