A381737 Orders k of Hermite polynomials whose maximal coefficient in absolute value appears twice.

8, 13, 34, 43, 76, 89, 134, 151, 208, 229, 298, 323, 404, 433, 526, 559, 664, 701, 818, 859, 988, 1033, 1174, 1223, 1376, 1429, 1594, 1651, 1828, 1889, 2078, 2143, 2344, 2413, 2626, 2699, 2924, 3001, 3238, 3319, 3568, 3653, 3914, 4003, 4276, 4369, 4654, 4751, 5048

Offset: 1

Author

Mike Sheppard, Mar 05 2025

Keywords

• nonn

Examples

			H_8(x) = 1680 - 13440 x^2 + 13440 x^4 - 3584 x^6 + 256 x^8, maximum coefficient in absolute value is 13440, which appears twice. Hence 8 is a term.
H_6(x) = -120 + 720 x^2 - 480 x^4 + 64 x^6. Absolute maximum unique. Hence 6 is not a term.
		

Links

• Mike Sheppard, Table of n, a(n) for n = 1..70

Crossrefs

Cf. A060821, A277281, A381524.

Programs

Mathematica

Flatten@Position[Table[Count[#, Max@#] &@Abs@CoefficientList[HermiteH[n, x], x], {n, 1000}], 2]

PARI

isok(k) = my(vp=apply(x->abs(x), Vec(polhermite(k))), m=vecmax(vp)); #select(x->(x==m), vp) == 2; \\ Michel Marcus, Mar 09 2025

Formula

Conjecture 1: a(n) = 2*n*(n + 2) + (n + 1)*(-1)^(n+1).

Conjecture 2: a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).

Conjecture 3: G.f.: (8*x + 5*x^2 + 5*x^3 - x^4 - x^5) / ((1 - x)^3 * (1 + x)^2).

Terms < 20000 consistent with conjectures. - Jinyuan Wang, Mar 09 2025.