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

%I A381737 #28 Mar 16 2025 18:11:54
%S A381737 8,13,34,43,76,89,134,151,208,229,298,323,404,433,526,559,664,701,818,
%T A381737 859,988,1033,1174,1223,1376,1429,1594,1651,1828,1889,2078,2143,2344,
%U A381737 2413,2626,2699,2924,3001,3238,3319,3568,3653,3914,4003,4276,4369,4654,4751,5048
%N A381737 Orders k of Hermite polynomials whose maximal coefficient in absolute value appears twice.
%H A381737 Mike Sheppard, <a href="/A381737/b381737.txt">Table of n, a(n) for n = 1..70</a>
%F A381737 Conjecture 1: a(n) = 2*n*(n + 2) + (n + 1)*(-1)^(n+1).
%F A381737 Conjecture 2: a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
%F A381737 Conjecture 3: G.f.: (8*x + 5*x^2 + 5*x^3 - x^4 - x^5) / ((1 - x)^3 * (1 + x)^2).
%F A381737 Terms < 20000 consistent with conjectures. - _Jinyuan Wang_, Mar 09 2025.
%e A381737 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.
%e A381737 H_6(x) = -120 + 720 x^2 - 480 x^4 + 64 x^6. Absolute maximum unique. Hence 6 is not a term.
%t A381737 Flatten@Position[Table[Count[#, Max@#] &@Abs@CoefficientList[HermiteH[n, x], x], {n, 1000}], 2]
%o A381737 (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
%Y A381737 Cf. A060821, A277281, A381524.
%K A381737 nonn
%O A381737 1,1
%A A381737 _Mike Sheppard_, Mar 05 2025