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.

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

Author

Roger L. Bagula, Jun 05 2008

Keywords

• sign
• tabf

Comments

Coefficients are ordered along increasing exponents [x^k], k=0,...,floor((n+1)/2).

Row sums are 2, 3, 4, 4, 4, -2, -8, -24, -40, -28, -16,..

Examples

			{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}.
		

Crossrefs

Cf. A060821.

Programs

Maple

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) ;

Mathematica

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}]

Formula

a(2*n,k) = 2* A060821(n,k). a(2*n-1,k) = A060821(n-1,k)+A060821(n,k) .

sum_{k=0..n} a(2*n,k) = 2*A062267(n).

sum_{k=0..n} a(2*n-1,k) = A062267(n) + A062267(n-1).

Extensions

Edited by the Associate Editors of the OEIS, Aug 28 2009