Known recursion link to Hermite polynomials not solved in Mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg69430] Known recursion link to Hermite polynomials not solved in Mathematica
• From: Roger Bagula <rlbagula at sbcglobal.net>
• Date: Tue, 12 Sep 2006 06:52:53 -0400 (EDT)

```A000898 in Sloan's OEIS:

a[0] = 1; a[1] = 2;
a[n_] := a[n] = 2*(a[n - 1] + (n - 1)*a[n - 2])
b = Table[a[n], {n, 0, 30}]

{1, 2, 6, 20, 76, 312, 1384, 6512, 32400, 168992, 921184, 5222208,
30710464,
186753920, 1171979904, 7573069568, 50305536256, 342949298688,
2396286830080,
17138748412928, 125336396368896, 936222729254912, 7136574106003456,
55466948299223040, 439216305474605056, 3540846129311916032,
29042507532354084864, 242209013788927803392, 2052713434324976189440,
17669131640829909368832, 154395642472508437725184}

Table[HermiteH[n, I]*I^n, {n, 0, 30}]

{1, -2, 6, -20, 76, -312, 1384, -6512,
32400, -168992, 921184, -5222208, 30710464, -186753920, 1171979904,
-7573069568, 50305536256, -342949298688, 2396286830080, -17138748412928,
125336396368896, -936222729254912,
7136574106003456, -55466948299223040, 439216305474605056,
-3540846129311916032, 29042507532354084864, -242209013788927803392,
2052713434324976189440, -17669131640829909368832, 154395642472508437725184}

Yet Mathematica refuses to give that solution ( using Bob Hanlon's solver):
f[n_Integer] = Module[{a}, a[n] /. RSolve[{a[n] == 2*( a[n - 1] + (n -
1)*a[n - 2]), a[0] == 1, a[1] == 2}, a[n], n][[1]] // FullSimplify]

My reason for wanting a solution is my own Hermite related recursion
A121966:
a[0] = 1; a[1] = 2;
a[n_] := a[n] = a[n - 1] - (n - 1)*a[n - 2]
Table[a[n], {n, 0, 30}]

{1, 2, 1, -3, -6, 6, 36, 0, -252, -252, 2016, 4536, -17640, -72072,
157248, 1166256, -1192464, -19852560, 419328, 357765408, 349798176,
-6805509984, -14151271680, 135569947968, 461049196608, -2792629554624,
-14318859469824, 58289508950400, 444898714635648, -1187207535975552,
-14089270260409344}

Which I think has some simple HermiteH type solution.

```

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