%I #6 Dec 08 2021 07:34:53
%S 0,1,-4,0,16,10,12,182,1120,7452,58640,520784,5142144,55929640,
%T 664505744,8562670920,118939979008,1771631324848,28168269788160,
%U 476151820931168,8526830353553920,161255217263900256
%N a(n) = n * n! * b(n), where b(n) = ((n-1)*(n-3)*b(n-1) - b(n-2) + b(n-3))/(n*(n - 1)) and b(0) = b(1) = 1, b(2) = -1.
%D Martin Braun, 'Differential Equations and Their Applications: An Introduction to Applied Mathematics', Texts in Applied Mathematics, Vol. 11, Springer, 4th ed., 1992, pp. 194-196, Example 5.
%H G. C. Greubel, <a href="/A158802/b158802.txt">Table of n, a(n) for n = 0..445</a>
%F a(n) = n * n! * b(n), where b(n) = ((n-1)*(n-3)*b(n-1) - b(n-2) + b(n-3))/(n*(n - 1)) and b(0) = b(1) = 1, b(2) = -1.
%F From _G. C. Greubel_, Dec 07 2021: (Start)
%F a(n) = (n/((n-1)*(n-2)*(n-3)))*((n-2)*(n-3)^2*a(n-1) - (n-1)*(n-3)*a(n-2) + (n-1)*(n-2)^2*a(n-3)), with a(0) = 0, a(1) = 1, a(2) = -4, a(3) = 0.
%F a(n) = n*n!*b(n), where y(t) = Sum_{n>=0} b(n)*t^n satisfies the differential equation (1-t)*y''(t) + y'(t) + (1-t)*y(t) = 0.
%F a(n) = n*n!*b(n), where y(t) = Sum_{n>=0} b(n)*t^n = (((1-t)*((BesselJ[0, 1] + 4*BesselJ[1, 1] - BesselJ[2, 1])*BesselY[1, t-1] - BesselJ[1, 1-t]*(BesselY[0, -1] - 4*BesselY[1, -1] - BesselY[2, -1])))/((BesselJ[0, 1] - BesselJ[2, 1]) BesselY[1, -1] + BesselJ[1, 1] (BesselY[0, -1] - BesselY[2, -1]))). (End)
%t b[0]:=1; b[1]:=1; b[2]:=-1;
%t b[n_]:= b[n]= ((n-1)*(n-3)*b[n-1] - b[n-2] + b[n-3])/(n*(n-1));
%t Table[n*n!*b[n], {n, 0, 30}]
%o (Sage)
%o @CachedFunction
%o def a(n):
%o if (n<3): return (-1)^(n+1)*n^2
%o elif (n==3): return 0
%o else: return (n/((n-1)*(n-2)*(n-3)))*((n-2)*(n-3)^2*a(n-1) -(n-1)*(n-3)*a(n-2) +(n-1)*(n-2)^2*a(n-3))
%o [a(n) for n in (0..30)] # _G. C. Greubel_, Dec 07 2021
%K sign
%O 0,3
%A _Roger L. Bagula_, Mar 27 2009
%E Edited by _G. C. Greubel_, Dec 07 2021
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