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%I #34 Sep 08 2022 08:46:17
%S 1,10,30,-300,-3850,-13940,56300,543400,-2332250,-29758500,340835780,
%T 7316239000,40381709500,-199606565000,-4494519345000,-25429880846000,
%U 18331676223750,848074482677500,714616060812500,-19019302889325000,506727569992188500
%N n^2*a(n) = 2*(14*n^2 - 16*n + 7)*a(n-1) - 20*(24*n^2 - 56*n + 41)*a(n-2) + 80*(64*n^2 - 224*n + 221)*a(n-3) - 1600*(24*n^2 - 112*n + 139)*a(n-4) + 6400*(28*n^2 - 164*n + 245)*a(n-5) - 128000*(2*n-7)^2*a(n-6) for n>6, a(0)=1, a(1)=10, a(2)=30, a(3)=-300, a(4)=-3850, a(5)=-13940, a(6) = 56300.
%H Gheorghe Coserea, <a href="/A276180/b276180.txt">Table of n, a(n) for n = 0..301</a>
%H Robert S. Maier, <a href="http://arxiv.org/abs/math/0611041">On Rationally Parametrized Modular Equations</a>, arXiv:math/0611041 [math.NT], 2006.
%F n^2*a(n) = 2*(14*n^2 - 16*n + 7)*a(n-1) - 20*(24*n^2 - 56*n + 41)*a(n-2) + 80*(64*n^2 - 224*n + 221)*a(n-3) - 1600*(24*n^2 - 112*n + 139)*a(n-4) + 6400*(28*n^2 - 164*n + 245)*a(n-5) - 128000*(2*n-7)^2*a(n-6) for n>6, a(0)=1, a(1)=10, a(2)=30, a(3)=-300, a(4)=-3850, a(5)=-13940, a(6)=56300.
%F 0 = 4*x*(x^2+2*x+5)*(x^4+5*x^3+15*x^2+25*x+25)*y'' + (24*x^6 + 144*x^5 + 520*x^4 + 1120*x^3 + 1600*x^2 + 1300*x + 500)*y' + 25*(x^5 + 5*x^4 + 15*x^3 + 25*x^2 + 25*x + 10)*y, where y(x) = A(x/-20).
%e A(x) = 1 + 10*x + 30*x^2 - 300*x^3 - 3850*x^4 - 13940*x^5 + ... is the g.f.
%t a[n_] := a[n] = If[n <= 6, {1, 10, 30, -300, -3850, -13940, 56300}[[n+1]], (1/n^2)(6400(28n^2 - 164n + 245) a[n-5] - 1600(24n^2 - 112n + 139) a[n-4] + 80(64n^2 - 224n + 221) a[n-3] - 20(24n^2 - 56n + 41) a[n-2] + 2(14n^2 - 16n + 7) a[n-1] - 128000(2n - 7)^2 a[n-6])];
%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Dec 16 2018 *)
%o (PARI)
%o seq(N) = {
%o my(a = vector(N));
%o a[1] = 10; a[2] = 30; a[3] = -300; a[4] = -3850; a[5] = -13940; a[6] = 56300;
%o for (n=7, N,
%o my(t1 = 2*(14*n^2 - 16*n + 7)*a[n-1] - 20*(24*n^2 - 56*n + 41)*a[n-2],
%o t2 = 80*(64*n^2 - 224*n + 221)*a[n-3] - 1600*(24*n^2 - 112*n + 139)*a[n-4],
%o t3 = 6400*(28*n^2 - 164*n + 245)*a[n-5] - 128000*(2*n-7)^2 * a[n-6]);
%o a[n] = (t1+t2+t3)/n^2);
%o concat(1,a);
%o };
%o seq(25)
%o (Magma) I:=[10,30,-300,-3850,-13940,56300]; [1] cat [n le 6 select I[n] else (2*(14*n^2-16*n+7)*Self(n-1)-20*(24*n^2- 56*n+41)*Self(n-2)+80*(64*n^2-224*n+221)*Self(n-3)-1600*(24*n^2-112*n+139)*Self(n-4)+6400*(28*n^2-164*n+245)*Self(n-5)-128000*(2*n-7)^2*Self(n-6))div n^2: n in [1..30]]; // _Vincenzo Librandi_, Aug 25 2016
%Y Cf. A091401, A276018.
%K sign
%O 0,2
%A _Gheorghe Coserea_, Aug 24 2016
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