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A276177 n^2 * a(n) = 6*(66*n^2 - 94*n + 41) * a(n-1) - 36*(2016*n^2 - 5712*n + 4387) * a(n-2) + 50544*(132*n^2 - 560*n + 609) * a(n-3) - 7884864*(6*n-17)^2*a(n-4), with a(0)=1, a(1)=78, a(2)=4446, a(3)=20124.

%I A276177 #24 Sep 08 2022 08:46:17
%S A276177 1,78,4446,20124,-38185290,-6138851004,-560711991060,-21068540562888,
%T A276177 3057536757534246,744702083933794740,85203074089262120004,
%U A276177 5052846560269468159368,-180318018879496001303748,-86176724948835065345458008,-11276003918572185562671306600,-751248675388448553292016521104
%N A276177 n^2 * a(n) = 6*(66*n^2 - 94*n + 41) * a(n-1) - 36*(2016*n^2 - 5712*n + 4387) * a(n-2) + 50544*(132*n^2 - 560*n + 609) * a(n-3) - 7884864*(6*n-17)^2*a(n-4), with a(0)=1, a(1)=78, a(2)=4446, a(3)=20124.
%H A276177 Gheorghe Coserea, <a href="/A276177/b276177.txt">Table of n, a(n) for n = 0..301</a>
%H A276177 Robert S. Maier, <a href="http://arxiv.org/abs/math/0611041">On Rationally Parametrized Modular Equations</a>, arXiv:math/0611041 [math.NT], 2006.
%F A276177 n^2 * a(n) = 6*(66*n^2 - 94*n + 41) * a(n-1) - 36*(2016*n^2 - 5712*n + 4387) * a(n-2) + 50544*(132*n^2 - 560*n + 609) * a(n-3) - 7884864*(6*n-17)^2*a(n-4), with a(0)=1, a(1)=78, a(2)=4446, a(3)=20124.
%F A276177 0 = 36*x*(x^2 + 5*x + 13)*(x^2 + 6*x + 13)*y'' + 12*(10*x^4 + 91*x^3 + 364*x^2 + 676*x + 507)*y' + (49*x^3 + 351*x^2 + 1027*x + 1014)*y, where y(x) = A(x/-468).
%e A276177 A(x) = 1 + 78*x + 4446*x^2 + 20124*x^3 + ... is the g.f.
%t A276177 a[0] = 1; a[1] = 78; a[2] = 4446; a[3] = 20124; a[n_] := a[n] = (6(66n^2 - 94n + 41) a[n-1] - 36(2016n^2 - 5712n + 4387)a[n-2] + 50544(132n^2 - 560n + 609)a[n-3] - 7884864(6n - 17)^2 a[n-4])/n^2;
%t A276177 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Oct 19 2018 *)
%o A276177 (PARI)
%o A276177 seq(N) = {
%o A276177   my(a = vector(N));
%o A276177   a[1] = 78; a[2] = 4446; a[3] = 20124; a[4] = -38185290;
%o A276177   for (n = 5, N,
%o A276177        my(t1 = 6*(66*n^2 - 94*n + 41) * a[n-1],
%o A276177           t2 = -36*(2016*n^2 - 5712*n + 4387) * a[n-2],
%o A276177           t3 = 50544*(132*n^2 - 560*n + 609) * a[n-3],
%o A276177           t4 = -7884864*(6*n-17)^2 * a[n-4]);
%o A276177        a[n] = (t1 + t2 + t3 + t4)/n^2);
%o A276177   concat(1,a);
%o A276177 };
%o A276177 seq(17)
%o A276177 (Magma) I:=[78,4446,20124,-38185290]; [1] cat [n le 4 select I[n] else (6*(66*n^2-94*n+41)*Self(n-1)-36*(2016*n^2-5712*n+4387)*Self(n-2)+50544*(132*n^2-560*n+ 609)*Self(n-3)-7884864*(6*n-17)^2*Self(n-4)) div n^2: n in [1..30]]; // _Vincenzo Librandi_, Aug 25 2016
%Y A276177 Cf. A091401, A276018.
%K A276177 sign
%O A276177 0,2
%A A276177 _Gheorghe Coserea_, Aug 23 2016