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A239164
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Number of partitions of 5^n into parts that are at most n with at least one part of each size.
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2
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0, 1, 12, 1240, 1655004, 32796849930, 10743023668660275, 62590747974586286694030, 6826987264035710020018176749475, 14471606032117455546329821353159274382372, 613427607589897771307393494301176209875530879140211
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OFFSET
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0,3
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..40
A. V. Sills and D. Zeilberger, Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz), arXiv:1108.4391 [math.CO], 2011.
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FORMULA
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a(n) = [x^(5^n-n*(n+1)/2)] Product_{j=1..n} 1/(1-x^j).
a(n) ~ 5^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015
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EXAMPLE
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a(2) = 12: 2222222222221, 22222222222111, 222222222211111, 2222222221111111, 22222222111111111, 222222211111111111, 2222221111111111111, 22222111111111111111, 222211111111111111111, 2221111111111111111111, 22111111111111111111111, 211111111111111111111111.
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MATHEMATICA
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maxExponent = 45; a[0] = 0; a[1] = 1;
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[5^n - n(n+1)/2 + 1] // Round];
Table[an = a[n]; Print[n, " ", an]; an, {n, 0, 10}] (* Jean-François Alcover, Nov 15 2018 *)
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CROSSREFS
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Column k=5 of A238012.
Sequence in context: A317953 A009155 A078296 * A209176 A137343 A133461
Adjacent sequences: A239161 A239162 A239163 * A239165 A239166 A239167
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KEYWORD
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nonn
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AUTHOR
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Alois P. Heinz, Mar 11 2014
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STATUS
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approved
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