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%I #4 Mar 30 2012 17:34:26
%S 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,
%T -96,0,32,12,120,-48,-160,16,32,0,240,0,-320,0,64,-120,120,720,-160,
%U -480,32,64,-240,0,1440,0,-960,0,128,-120,-1680,720,3360,-480,-1344,64,128,0,-3360,0,6720,0,-2688
%N 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.
%C Coefficients are ordered along increasing exponents [x^k], k=0,...,floor((n+1)/2).
%C Row sums are 2, 3, 4, 4, 4, -2, -8, -24, -40, -28, -16,..
%F a(2*n,k) = 2* A060821(n,k). a(2*n-1,k) = A060821(n-1,k)+A060821(n,k) .
%F sum_{k=0..n} a(2*n,k) = 2*A062267(n).
%F sum_{k=0..n} a(2*n-1,k) = A062267(n) + A062267(n-1).
%e {2}, = 2
%e {1, 2}, = 1+2x
%e {0, 4}, = 4x^2
%e {-2, 2, 4}, = -2+2x+4x^2
%e {-4, 0, 8}, = -4+8x^2
%e {-2, -12, 4, 8},
%e {0, -24, 0, 16},
%e {12, -12, -48, 8, 16},
%e {24, 0, -96, 0, 32},
%e {12, 120, -48, -160, 16, 32},
%e {0, 240, 0, -320, 0, 64}.
%p A060821 := proc(n,k) orthopoly[H](n,x) ; coeftayl(%,x=0,k) ; end:
%p 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) ;
%t 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]];
%t Table[ExpandAll[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}];
%t Flatten[a] Table[Apply[Plus, CoefficientList[p[x, n], x]], {n, 0, 10}]
%Y Cf. A060821.
%K sign,tabf
%O 0,1
%A _Roger L. Bagula_, Jun 05 2008
%E Edited by the Associate Editors of the OEIS, Aug 28 2009
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