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A117144
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Partitions of n in which each part k occurs at least k times.
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22
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1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 8, 9, 10, 12, 15, 16, 19, 21, 25, 28, 32, 34, 41, 46, 51, 55, 64, 70, 79, 86, 97, 106, 119, 129, 146, 159, 175, 190, 214, 232, 256, 277, 306, 334, 367, 394, 434, 472, 515, 556, 607, 654, 714, 770, 836, 901, 978, 1048, 1140, 1226, 1322
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OFFSET
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0,5
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COMMENTS
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The Heinz numbers of these integer partitions are given by A324525. - Gus Wiseman, Mar 09 2019
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LINKS
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FORMULA
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G.f.: Product_{k>=1} (1-x^k+x^(k^2))/(1-x^k).
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EXAMPLE
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a(9)=5 because we have [3,3,3], [2,2,2,2,1], [2,2,2,1,1,1], [2,2,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1].
The a(1) = 1 through a(9) = 5 integer partitions:
1 11 111 22 221 222 2221 2222 333
1111 11111 2211 22111 22211 22221
111111 1111111 221111 222111
11111111 2211111
111111111
(End)
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MAPLE
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g:=product((1-x^k+x^(k^2))/(1-x^k), k=1..100): gser:=series(g, x=0, 70): seq(coeff(gser, x, n), n=0..66);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1) +add(b(n-i*j, i-1), j=i..n/i)))
end:
a:= n-> b(n$2):
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + Sum[b[n-i*j, i-1], {j, i, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Feb 03 2017, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n], And@@Table[Count[#, i]>=i, {i, Union[#]}]&]], {n, 0, 30}] (* Gus Wiseman, Mar 09 2019 *)
nmax = 100; CoefficientList[Series[Product[(1-x^k+x^(k^2))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 28 2024 *)
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CROSSREFS
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Cf. A001462, A003114, A006141, A033461, A039900, A047993, A052335, A064174, A090858, A114638, A115584, A276078, A280204.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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