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A238001 Number of partitions of n^n into parts that are at most n with at least one part of each size. 5
0, 1, 1, 48, 109809, 32796849930, 2555847904495965819, 85962759806610904434664386174, 1841132100297745277187328924904656111127054, 34687813181057391872792859998288408847592250236051615502024 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..27

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.

FORMULA

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

a(n) ~ n^(n*(n-1)) / (n!*(n-1)!) ~ exp(2*n) * n^(n*(n-3)) / (2*Pi). - Vaclav Kotesovec, Jun 05 2015

EXAMPLE

a(1) = 1: 1.

a(2) = 1: 211.

a(3) = 48: 3333333321, ..., 321111111111111111111111.

MATHEMATICA

maxExponent = 50; 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[n^n-n(n+1)/2 + 1] // Round];

Table[an = a[n]; Print[n, " ", an]; an, {n, 0, 9}] (* Jean-Fran├žois Alcover, Nov 15 2018 *)

CROSSREFS

Main diagonal of A238012.

Cf. A236810, A237512, A237998, A237999, A238000.

Sequence in context: A006070 A081262 A340186 * A228143 A008704 A037947

Adjacent sequences:  A237998 A237999 A238000 * A238002 A238003 A238004

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Feb 16 2014

STATUS

approved

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Last modified January 17 21:35 EST 2021. Contains 340247 sequences. (Running on oeis4.)