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%I #19 Sep 08 2022 08:46:17
%S 1,6,24,78,216,504,906,756,-2808,-17832,-57312,-104832,81882,1734156,
%T 9360576,35755956,106475472,232967664,215497680,-1178534304,
%U -8734303296,-36146763648,-108833048064,-220247838720,-46688571558,2220777704700,13473296923536,53523581091900
%N n^2 * a(n) = 2*(7*n^2 - 7*n + 3)*a(n-1) - 12*(7*n^2 - 14*n + 9)*a(n-2) + 39*(7*n^2 - 21*n + 18) * a(n-3) - 72*(7*n^2 - 28*n + 30)*a(n-4) + 72*(7*n^2 - 35*n + 45) * a(n-5) - 216*(n-3)^2 * a(n-6), with a(0)=1, a(1)=6, a(2)=24, a(3)=78, a(4)=216, a(5)=504.
%H Gheorghe Coserea, <a href="/A276179/b276179.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*(7*n^2 - 7*n + 3)*a(n-1) - 12*(7*n^2 - 14*n + 9)*a(n-2) + 39*(7*n^2 - 21*n + 18) * a(n-3) - 72*(7*n^2 - 28*n + 30)*a(n-4) + 72*(7*n^2 - 35*n + 45) * a(n-5) - 216*(n-3)^2 * a(n-6), with a(0)=1, a(1)=6, a(2)=24, a(3)=78, a(4)=216, a(5)=504.
%F 0 = x*(x+2)*(x+3)*(x^2+3*x+3)*(x^2+6*x+12)*y'' + (7*x^6 + 84*x^5 + 420*x^4 + 1092*x^3 + 1512*x^2 + 1008*x + 216)*y' + 9*(x+2)^2 * (x^3 + 6*x^2 + 12*x + 6)*y, where y(x) = A(x/-6).
%e A(x) = 1 + 6*x + 24*x^2 + 78*x^3 + 216*x^4 + 504*x^5 + 906*x^6 + ... is the g.f.
%o (PARI)
%o seq(N) = {
%o my(a = vector(N));
%o a[1] = 6; a[2] = 24; a[3] = 78; a[4] = 216; a[5] = 504; a[6] = 906;
%o for (n = 7, N,
%o my(t1 = 2*(7*n^2 - 7*n + 3)*a[n-1] - 12*(7*n^2 - 14*n + 9)*a[n-2],
%o t2 = 39*(7*n^2 - 21*n + 18) * a[n-3] - 72*(7*n^2 - 28*n + 30)*a[n-4],
%o t3 = 72*(7*n^2 - 35*n + 45) * a[n-5] - 216*(n-3)^2 * a[n-6]);
%o a[n] = (t1+t2+t3)/n^2);
%o concat(1,a);
%o };
%o seq(33)
%o (Magma) I:=[6,24,78,216,504,906]; [1] cat [n le 6 select I[n] else (2*(7*n^2-7*n+3)*Self(n-1)-12*(7*n^2-14*n+9)*Self(n-2)+39*(7*n^2-21*n+18)*Self(n-3)-72*(7*n^2-28*n+30)*Self(n-4)+72*(7*n^2-35*n+45)*Self(n-5)-216*(n-3)^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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