%I #25 Feb 02 2024 07:43:46

%S 1,1,2,2,2,3,3,4,6,7,10,15,21,33,50,83,126,208,321,498,688,934,1208,

%T 1496,1798,2140,2482,2862,3268,3690,4145,4619,5142,5687,6265,6880,

%U 7530,8214,8937,9700,10489,11339,12218,13142,14112,15123,16181,17288,18438,19639,20888

%N Number of different coefficient values in expansion of Product_{k=1..n} (1+x^(k^2)).

%H Vaclav Kotesovec, <a href="/A369786/b369786.txt">Table of n, a(n) for n = 0..500</a>

%F Conjecture: a(n) ~ n^3/6. - _Vaclav Kotesovec_, Feb 02 2024

%t p = 1; Join[{1}, Table[p = Expand[p*(1 + x^(n^2))]; Length[Union[CoefficientList[p, x]]], {n, 1, 50}]] (* or *)

%t nmax = 50; poly = ConstantArray[0, nmax*(nmax + 1)*(2*nmax + 1)/6 + 1]; poly[[1]] = 1; poly[[2]] = 1; Flatten[{{1, 1}, Table[Do[poly[[j + 1]] += poly[[j - k^2 + 1]], {j, k*(k + 1)*(2*k + 1)/6, k^2, -1}]; Length[Union[poly]], {k, 2, nmax}]}] (* _Vaclav Kotesovec_, Feb 01 2024 *)

%o (PARI) a(n) = #Set(Vec(prod(k=1, n, 1+x^k^2)));

%o (Python)

%o from collections import Counter

%o def A369786(n):

%o c = {0:1}

%o for k in range(1,n+1):

%o m, d = k**2, Counter(c)

%o for j in c:

%o d[j+m] += c[j]

%o c = d

%o return len(set(c.values()))+int(max(c)+1>len(c)) # _Chai Wah Wu_, Feb 01 2024

%Y Cf. A033461, A039822.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Feb 01 2024