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A239162 Number of partitions of 3^n into parts that are at most n with at least one part of each size. 2

%I

%S 0,1,4,48,3042,1067474,2215932130,29012104252380,2526293243761311036,

%T 1525710603023191548743988,6600334932211428773703751221040,

%U 209705652574790086852527310591449309624,49907101066058865036206450041083799915221352487

%N Number of partitions of 3^n into parts that are at most n with at least one part of each size.

%H Alois P. Heinz, <a href="/A239162/b239162.txt">Table of n, a(n) for n = 0..49</a>

%H A. V. Sills and D. Zeilberger, <a href="https://arxiv.org/abs/1108.4391">Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz)</a>, arXiv:1108.4391 [math.CO], 2011.

%F a(n) = [x^(3^n-n*(n+1)/2)] Product_{j=1..n} 1/(1-x^j).

%F a(n) ~ 3^(n*(n-1)) / (n!*(n-1)!). - _Vaclav Kotesovec_, Jun 05 2015

%e a(2) = 4: 22221, 222111, 2211111, 21111111.

%t maxExponent = 30; a[0] = 0; a[1] = 1;

%t a[n_] := Module[{}, aparts = List @@ (Product[1/(1 - x^j), {j, 1, n}] // Apart); cc = aparts + O[x]^maxExponent // CoefficientList[#, x]&; f[k_] = Total[FindSequenceFunction[#, k]& /@ cc]; f[3^n - n(n+1)/2 + 1] // Round]; Table[an = a[n];

%t Print[n, " ", an]; an, {n, 0, 12}] (* _Jean-Fran├žois Alcover_, Nov 15 2018 *)

%Y Column k=3 of A238012.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Mar 11 2014

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Last modified August 8 14:59 EDT 2022. Contains 356009 sequences. (Running on oeis4.)