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A239163 Number of partitions of 4^n into parts that are at most n with at least one part of each size. 2
0, 1, 7, 310, 109809, 370702459, 13173778523786, 5303087097326728307, 25501946239758780918956349, 1523132187565775833398409415522245, 1163511401871888391788752975911167467265905, 11631778554448496258128131825307023131265496068454454 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..43

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^(4^n-n*(n+1)/2)] Product_{j=1..n} 1/(1-x^j).

a(n) ~ 4^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015

EXAMPLE

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

MATHEMATICA

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

Table[an = a[n]; Print[n, " ", an]; an, {n, 0, 11}] (* Jean-Fran├žois Alcover, Nov 15 2018 *)

CROSSREFS

Column k=4 of A238012.

Sequence in context: A257919 A002437 A300870 * A086215 A119163 A171148

Adjacent sequences:  A239160 A239161 A239162 * A239164 A239165 A239166

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Mar 11 2014

STATUS

approved

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Last modified February 19 13:03 EST 2020. Contains 332044 sequences. (Running on oeis4.)