linear secod order homogeneous differential equation recursions

• To: mathgroup at smc.vnet.net
• Subject: [mg69852] linear secod order homogeneous differential equation recursions
• From: Roger Bagula <rlbagula at sbcglobal.net>
• Date: Mon, 25 Sep 2006 03:53:29 -0400 (EDT)

```Again I have a difficult problem.
I have this factorial based recursion:
a[n] = (a0*n^2 + b0*n + c0)*a[n - 2]/(n*(n - 1))

a[n]*n!=Integer
I want to get a form:
b[n]=a[n]*n!
I've tried your RSolve , but it doesn't give a form out I can use.

M athematica:
Clear[a, a0, b0, c0]
0 = 1; b0 = -2; c0 = -1;
a[n_] := a[n] = (a0*n^2 + b0*n + c0)*a[n
- 2]/(n*(n - 1)); a[0] = 1; a[1] = 1;
Table[ExpandAll[a[n]*n!], {n, 0, 30}]
{1, 1, -1, 2, -7, 28, -161,
952, -7567, 59024, -597793, 5784352, -71137367, 821377984, -11879940289,
159347328896, -2649226684447,
40474221539584, -760328058436289, 13032699335746048, -272957772978627751,
5187014335626927104, -119828462337617582689, 2500140909772178864128,
-63149599651924466077103, 1435080882209230668009472,
-39342200583148942366035169, 967244514609021470238384128,
-28601779823949281100107567863, 756385210424254789726416388096,
-23996893272293446842990249437057}

Clear[a]
a[n_] := a[n] = -(n^2 - n - 1)*a[n
- 2]/(n*(n - 1)); a[0] = 1; a[1] = 1;
Table[a[n]*n!, {n, 0, 30}]
{1, 1, -1, -5,
11, 95, -319, -3895, 17545, 276545, -1561505, -30143405, 204557155,
4672227775, -37024845055, -976495604975, 8848937968145, 264630308948225,
-2698926080284225, -90238935351344725, 1022892984427721275,
37810113912213439775, -471553665821179507775, -19094107525667787086375,
259826069867469908784025, 11437370407875004464738625,
-168627119343987970800832225, -8017596655920378129781776125,
127313475104710917954628329875, 6502270887951426663253020437375,
-110635409865993787702572018661375}

Clear[a]
a[n_] := a[n] = (n - 2)*a[n
- 2]/(n*(n - 1)); a[0] = 1; a[1] = 1;
Table[a[n]*n!, {n, 0, 30}]
{1, 1, 0, 1, 0, 3, 0, 15, 0, 105, 0, 945, 0, 10395, 0, 135135, 0,
2027025, 0,
34459425, 0, 654729075, 0, 13749310575, 0, 316234143225, 0,
7905853580625, 0,
213458046676875, 0}

I didn't find these on my own:
they were recursion examples in an old  differential equation text
I had. So far none of my results as a[n]*n! haven't been in OEIS.
These are solution to differtential equations of the type:
y''+p(x)*y'+q(x)*y=0
f(x)=y=Sum[a[n]*(x-x0)^n,{n,0,Infinity}]
So in Taylor series terms I'm getting ( I think):
b[n]=D[f(x0),{x,n}]
a[n]=b[n]/n!

I need a gerenal form for the derivatives b[n].

```

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