This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A381524 #18 Mar 18 2025 07:22:26
%S A381524 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,
%T A381524 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,
%U A381524 11,10,11,10,11,10,11,10,11,10,11,10,11,12
%N A381524 Smallest exponent of x of maximal coefficient (ignoring signs) in Hermite polynomial of order n.
%C A381524 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.
%C A381524 Conjecture: Differences are either +1 or -1.
%e A381524 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.
%e A381524 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.
%t A381524 Table[(PositionLargest@Abs[CoefficientList[HermiteH[n, x], x]])[[1]] - 1, {n, 0, 100}]
%o A381524 (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
%Y A381524 Cf. A277281 (maximal coefficient ignoring signs), A381737 (non-unique exponents).
%K A381524 nonn
%O A381524 0,3
%A A381524 _Mike Sheppard_, Feb 26 2025