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%I #16 Sep 08 2022 08:46:15
%S 1,1,0,0,2,8,38,230,1620,12996,117184,1173424,12920440,155160880,
%T 2018252088,28268334072,424179011312,6788869628240,115438898013920,
%U 2078322337814624,39494885173772256,790012720199894976,16592338685780314720,365070830955616246240
%N a(n) = (n-1)*a(n-1) - a(n-2) + (n-2)*a(n-3) with a(0)=a(1)=1, a(2)=0.
%H Alois P. Heinz, <a href="/A269509/b269509.txt">Table of n, a(n) for n = 0..450</a>
%F E.g.f. A(x) satisfies 0 = A'(x)*(A'(x) + A'''(x)) - (A(x) + A''(x))*(A(x) + 2*A''(x)).
%F 0 = a(n)*(+a(n+1) - a(n+2) + 2*a(n+3) - a(n+4)) + a(n+1)*(+a(n+1) + 2*a(n+3)) + a(n+2)*(-a(n+2) + a(n+3) - a(n+4)) + a(n+3)*(+a(n+3)) for all n>=0.
%F a(n) ~ c * (n-1)!, where c = BesselJ(0,1) - BesselJ(1,1) = 0.3251471008130330354900353223837483... . - _Vaclav Kotesovec_, Feb 29 2016
%e G.f. = 1 + x + 2*x^4 + 8*x^5 + 38*x^6 + 230*x^7 + 1620*x^8 + 12996*x^9 + ...
%t RecurrenceTable[{a[n] == (n-1)*a[n-1] - a[n-2] + (n-2)*a[n-3], a[0]==a[1]==1, a[2]==0}, a, {n,0,20}] (* _Vaclav Kotesovec_, Mar 07 2016 *)
%t FullSimplify[Assuming[Element[x, Reals], CoefficientList[Series[Pi/2 * ((BesselJ[1, 1] - BesselJ[0, 1]) * BesselY[0, -1 + x] + BesselJ[0, 1 - x] * (BesselY[0, -1] + BesselY[1, -1])), {x, 0, 20}], x] * Range[0, 20]!]] (* _Vaclav Kotesovec_, Mar 07 2016 *)
%o (PARI) {a(n) = if( n<3, n>=0&&n<2, (n-1)*a(n-1) - a(n-2) + (n-2)*a(n-3))};
%o (PARI) {a(n) = my(A); if( n<3, n>=0&&n<2, A = vector(n, i); A[1]=1; for(i=4, n, A[i] = (i-1)*A[i-1] - A[i-2] + (i-2)*A[i-3]); A[n])};
%o (Magma) I:=[1,0,0]; [1] cat [n le 3 select I[n] else (n-1)*Self(n-1) - Self(n-2) + (n - 2)*Self(n-3): n in [1..50]]; // _G. C. Greubel_, Jul 30 2018
%Y Cf. A269450, A269820.
%K nonn,easy
%O 0,5
%A _Michael Somos_, Feb 28 2016
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