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Number of partitions of n into divisors of n where every part appears at least 2 times.
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%I #13 Jan 22 2025 22:26:45

%S 0,1,1,2,1,4,1,5,3,6,1,15,1,8,7,16,1,29,1,32,9,12,1,104,5,14,13,60,1,

%T 171,1,81,15,18,14,448,1,20,17,326,1,426,1,147,99,24,1,1675,7,173,23,

%U 205,1,902,23,809,25,30,1,8616,1,32,183,682,27,1629,1,354,31,1309

%N Number of partitions of n into divisors of n where every part appears at least 2 times.

%H Antti Karttunen, <a href="/A339243/b339243.txt">Table of n, a(n) for n = 1..10401</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F a(n) = [x^n] Product_{d|n} (1 + x^(2*d) / (1 - x^d)).

%e a(15) = 7 because we have [5, 5, 5], [3, 3, 3, 3, 3], [5, 5, 1, 1, 1, 1, 1], [3, 3, 3, 3, 1, 1, 1], [3, 3, 3, 1, 1, 1, 1, 1, 1], [3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

%t Table[SeriesCoefficient[Product[1 + x^(2 d)/(1 - x^d), {d, Divisors[n]}], {x, 0, n}], {n, 1, 70}]

%o (PARI) A339243(n) = { my(p=1); fordiv(n, d, p *= (1 + 'x^(2*d) / (1 - 'x^d))); polcoeff(Ser(p, 'x, 1+n), n); }; \\ _Antti Karttunen_, Jan 22 2025

%Y Cf. A000040 (positions of 1's), A007690, A018818, A027750.

%K nonn,changed

%O 1,4

%A _Ilya Gutkovskiy_, Nov 28 2020