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.