%I #15 Nov 15 2018 11:10:31

%S 0,1,7,310,109809,370702459,13173778523786,5303087097326728307,

%T 25501946239758780918956349,1523132187565775833398409415522245,

%U 1163511401871888391788752975911167467265905,11631778554448496258128131825307023131265496068454454

%N Number of partitions of 4^n into parts that are at most n with at least one part of each size.

%H Alois P. Heinz, <a href="/A239163/b239163.txt">Table of n, a(n) for n = 0..43</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^(4^n-n*(n+1)/2)] Product_{j=1..n} 1/(1-x^j).

%F a(n) ~ 4^(n*(n-1)) / (n!*(n-1)!). - _Vaclav Kotesovec_, Jun 05 2015

%e a(2) = 7: 222222211, 2222221111, 22222111111, 222211111111, 2221111111111, 22111111111111, 211111111111111.

%t maxExponent = 40; 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[4^n-n(n+1)/2 + 1] // Round];

%t Table[an = a[n]; Print[n, " ", an]; an, {n, 0, 11}] (* _Jean-François Alcover_, Nov 15 2018 *)

%Y Column k=4 of A238012.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Mar 11 2014