A276022 - OEIS

%I #18 Sep 08 2022 08:46:17

%S 1,6,30,144,690,3348,16536,83232,426618,2223180,11756052,62959680,

%T 340881792,1862954784,10262937600,56926831104,317632207194,

%U 1781352834300,10034760283356,56748881420640,322033934657628,1833043230774360,10462349697348528,59861990921495616

%N n^2 * a(n) = 3*(5*n^2 - 5*n + 2) * a(n-1) - 16*(5*n^2 - 10*n + 6) * a(n-2) + 36*(5*n^2 - 15*n + 12) * a(n-3) - 144*(n-2)^2 * a(n-4), with a(0)=1, a(1)=6, a(2)=30, a(3)=144.

%H Gheorghe Coserea, <a href="/A276022/b276022.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) = 3*(5*n^2 - 5*n + 2) * a(n-1) - 16*(5*n^2 - 10*n + 6) * a(n-2) + 36*(5*n^2 - 15*n + 12) * a(n-3) - 144*(n-2)^2 * a(n-4), with a(0)=1, a(1)=6, a(2)=30, a(3)=144.

%F 0 = x*(x+2)*(x+3)*(x+4)*(x+6)*y'' + (5*x^4 + 60*x^3 + 240*x^2 + 360*x + 144)*y' + 4*(x^2+6*x+6)*(x+3)*y, where y(x) = A(x/-12).

%F a(n) ~ 2^n * 3^(n+3/2) / (Pi*n). - _Vaclav Kotesovec_, Aug 25 2016

%e A(x) = 1 + 6*x + 30*x^2 + 144*x^3 + ... is the g.f.

%t a[0] = 1; a[1] = 6; a[2] = 30; a[3] = 144; a[n_] := a[n] = (3(5n^2 - 5n + 2) a[n-1] - 16(5n^2 - 10n + 6)a[n-2] + 36(5n^2 - 15n + 12) a[n-3] - 144(n-2)^2 a[n-4])/n^2;

%t Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Oct 19 2018 *)

%o (PARI)

%o seq(N) = {

%o   my(a = vector(N)); a[1] = 6; a[2] = 30; a[3] = 144; a[4] = 690;

%o   for (n=5, N,

%o        my(t1 = 3*(5*n^2 - 5*n + 2)*a[n-1] - 16*(5*n^2 - 10*n + 6)*a[n-2],

%o           t2 = 36*(5*n^2 - 15*n + 12)*a[n-3] - 144*(n-2)^2 * a[n-4]);

%o        a[n] = (t1+t2)/n^2);

%o   concat(1,a);

%o };

%o seq(25)

%o (Magma) I:=[6,30,144,690]; [1] cat [n le 4 select I[n] else (3*(5*n^2-5*n+2)*Self(n-1)-16*(5*n^2-10*n+6)*Self(n-2)+36*(5*n^2-15*n+12)*Self(n-3)-144*(n-2)^2*Self(n-4)) div n^2: n in [1..30]]; // _Vincenzo Librandi_, Aug 25 2016

%Y Cf. A091401, A276018.

%K nonn

%O 0,2

%A _Gheorghe Coserea_, Aug 23 2016