A381524 Smallest exponent of x of maximal coefficient (ignoring signs) in Hermite polynomial of order n.

0, 1, 2, 1, 2, 3, 2, 3, 2, 3, 4, 3, 4, 3, 4, 5, 4, 5, 4, 5, 6, 5, 6, 5, 6, 5, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 10, 9, 10, 9, 10, 9, 10, 9, 10, 9, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 12

Offset: 0

Author

Mike Sheppard, Feb 26 2025

Keywords

• nonn

Comments

Exponent is unique except for order of n within A381737, whose maximum of absolute value of coefficients appear twice. For example, 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. For those values a(n) and a(n)+2 both are maximums, in absolute value.

Conjecture: Differences are either +1 or -1.

Examples

			For n = 5, H_5(x) = 32*x^5 - 160*x^3 + 120*x. The maximal coefficient (ignoring signs) is 160, occurring at x^3, hence a(5) = 3.
For n = 8, H_8(x) = 1680 - 13440 x^2 + 13440 x^4 - 3584 x^6 + 256 x^8. The maximal coefficient (ignoring signs) is 13440, occurring at both x^2 and x^4, the smallest exponent being 2, hence a(8) = 2.
		

Crossrefs

Cf. A277281 (maximal coefficient ignoring signs), A381737 (non-unique exponents).

Programs

Mathematica

Table[(PositionLargest@Abs[CoefficientList[HermiteH[n, x], x]])[[1]] - 1, {n, 0, 100}]

PARI

a(n) = my(p=polhermite(n), m=vecmax(apply(x->abs(x), Vec(p)))); for(i=0, poldegree(p), if (abs(polcoef(p, i)) == m, return(i))); \\ Michel Marcus, Feb 26 2025