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 A381423 #8 Mar 06 2025 12:00:16
%S A381423 0,1,2,3,4,1,2,3,4,5,2,3,4,5,6,3,4,5,6,7,4,5,6,7,4,5,6,7,8,5,6,7,8,5,
%T A381423 6,7,8,9,6,7,8,9,6,7,8,9,10,7,8,9,10,7,8,9,10,11,8,9,10,11,8,9,10,11,
%U A381423 12,9,10,11,12,9,10,11,12,9,10,11,12,13,10
%N A381423 Exponent of x of maximal coefficient in Hermite polynomial of order n.
%C A381423 The exponent is always unique. The coefficients, in absolute value, follow a unimodal pattern, and their signs alternate. If the maximum absolute coefficient appears twice due to symmetry (e.g., H_8(x)), the terms will have opposite signs, ensuring a unique exponent for the maximum signed coefficient.
%C A381423 Conjecture: Differences are either 1 or -3; more specifically the patterns (1,1,1,-3) or (1,1,1,1,-3), with position of those patterns appearing at linearly and quadratically spaced intervals, respectively. Seems to grow O(n^(1/2))
%e A381423 For n = 5, H_5(x) = 32*x^5 - 160*x^3 + 120*x. The maximal coefficient is 120 (we take signs into account, so -160 < 120), occurring at x^1, hence a(5) = 1.
%t A381423 Table[(PositionLargest@CoefficientList[HermiteH[n, x], x])[[1]] - 1, {n, 0, 100}]
%o A381423 (PARI) a(n) = my(p=polhermite(n), m=vecmax(Vec(p))); for(i=0, poldegree(p), if (polcoef(p, i) == m, return(i))); \\ _Michel Marcus_, Feb 23 2025
%Y A381423 Cf. A277280 (maximal coefficient).
%K A381423 nonn
%O A381423 0,3
%A A381423 _Mike Sheppard_, Feb 23 2025