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Maximal coefficient of (1 + x) * (1 + x^3) * (1 + x^6) * ... * (1 + x^(n*(n+1)/2)).
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%I #16 Dec 29 2022 03:04:34

%S 1,1,1,1,2,2,3,4,5,7,12,18,27,44,73,122,210,362,620,1050,1857,3290,

%T 5949,10665,19086,34330,62252,113643,209460,383888,706457,1300198,

%U 2407535,4468367,8331820,15525814,28987902,54180854,101560631,190708871,358969426

%N Maximal coefficient of (1 + x) * (1 + x^3) * (1 + x^6) * ... * (1 + x^(n*(n+1)/2)).

%H Seiichi Manyama, <a href="/A359348/b359348.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) ~ sqrt(5) * 2^(n + 3/2) / (sqrt(Pi) * n^(5/2)). - _Vaclav Kotesovec_, Dec 29 2022

%e (1 + x) * (1 + x^3) * (1 + x^6) * (1 + x^10) = 1 + x + x^3 + x^4 + x^6 + x^7 + x^9 + 2 * x^10 + x^11 + x^13 + x^14 + x^16 + x^17 + x^19 + x^20. So a(4) = 2.

%o (PARI) a(n) = vecmax(Vec(prod(k=1, n, 1+x^(k*(k+1)/2))));

%Y Cf. A000217, A024940, A025591, A158380, A160235.

%K nonn

%O 0,5

%A _Seiichi Manyama_, Dec 27 2022