A122031
a(n) = a(n - 1) + (n - 1)*a(n - 2).
Original entry on oeis.org
1, 2, 3, 7, 16, 44, 124, 388, 1256, 4360, 15664, 59264, 231568, 942736, 3953120, 17151424, 76448224, 350871008, 1650490816, 7966168960, 39325494464, 198648873664, 1024484257408, 5394759478016, 28957897398400
Offset: 0
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a[0] = 1; a[1] = 2; a[n_] := a[n] = a[n - 1] + (n - 1)*a[n - 2] Table[a[n], {n, 0, 30}]
Table[n!*SeriesCoefficient[1/2*Exp[x+x^2/2]*(2-Sqrt[2*E*Pi]*Erf[1/Sqrt[2]]+Sqrt[2*E*Pi]*Erf[(1+x)/Sqrt[2]]),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec after Paul Abbott, Dec 27 2012 *)
RecurrenceTable[{a[0]==1,a[1]==2,a[n]==a[n-1]+(n-1)a[n-2]},a,{n,30}] (* Harvey P. Dale, Feb 21 2015 *)
A139158
Triangle a(n,k) of the expansion coefficients of the Hermite polynomial 2*H(n/2,x) if n even, of H((n-1)/2,x)+H((n+1)/2,x) if n odd.
Original entry on oeis.org
2, 1, 2, 0, 4, -2, 2, 4, -4, 0, 8, -2, -12, 4, 8, 0, -24, 0, 16, 12, -12, -48, 8, 16, 24, 0, -96, 0, 32, 12, 120, -48, -160, 16, 32, 0, 240, 0, -320, 0, 64, -120, 120, 720, -160, -480, 32, 64, -240, 0, 1440, 0, -960, 0, 128, -120, -1680, 720, 3360, -480, -1344, 64, 128, 0, -3360, 0, 6720, 0, -2688
Offset: 0
{2}, = 2
{1, 2}, = 1+2x
{0, 4}, = 4x^2
{-2, 2, 4}, = -2+2x+4x^2
{-4, 0, 8}, = -4+8x^2
{-2, -12, 4, 8},
{0, -24, 0, 16},
{12, -12, -48, 8, 16},
{24, 0, -96, 0, 32},
{12, 120, -48, -160, 16, 32},
{0, 240, 0, -320, 0, 64}.
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A060821 := proc(n,k) orthopoly[H](n,x) ; coeftayl(%,x=0,k) ; end:
A139158 := proc(n,k) if type(n,'even') then 2*A060821(n/2,k) ; else A060821((n+1)/2-1,k)+A060821((n+1)/2,k) ; fi; end: seq( seq(A139158(n,k),k=0..(n+1)/2),n=0..15) ;
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Clear[p, x] p[x, 0] = 2*HermiteH[0, x]; p[x, 1] = HermiteH[0, x] + HermiteH[1, x]; p[x, 2] = 2*HermiteH[1, x]; p[x_, m_] := p[x, m] = If[Mod[m, 2] == 0, 2*HermiteH[Floor[m/2], x], HermiteH[ Floor[m/2], x] + HermiteH[Floor[m/ 2 + 1], x]];
Table[ExpandAll[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}];
Flatten[a] Table[Apply[Plus, CoefficientList[p[x, n], x]], {n, 0, 10}]
Edited by the Associate Editors of the OEIS, Aug 28 2009
A277379
E.g.f.: exp(x/(1-x^2))/sqrt(1-x^2).
Original entry on oeis.org
1, 1, 2, 10, 40, 296, 1936, 17872, 164480, 1820800, 21442816, 279255296, 3967316992, 59837670400, 988024924160, 17009993230336, 318566665977856, 6177885274406912, 129053377688043520, 2786107670662021120, 64136976817284448256, 1525720008470138454016
Offset: 0
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Table[Abs[HermiteH[n, (1 + I)/2]]^2/2^n, {n, 0, 20}]
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