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A276429
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Number of partitions of n containing no part i of multiplicity i.
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18
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1, 0, 2, 2, 3, 5, 8, 9, 16, 19, 29, 36, 53, 65, 92, 115, 154, 195, 257, 318, 419, 516, 663, 821, 1039, 1277, 1606, 1963, 2441, 2978, 3675, 4454, 5469, 6603, 8043, 9688, 11732, 14066, 16963, 20260, 24310, 28953, 34586, 41047, 48857, 57802, 68528, 80862, 95534, 112388, 132391
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OFFSET
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0,3
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COMMENTS
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The Heinz numbers of these partitions are given by A325130. - Gus Wiseman, Apr 02 2019
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LINKS
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Vaclav Kotesovec, Table of n, a(n) for n = 0..12782 (terms 0..5000 from Alois P. Heinz)
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FORMULA
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a(n) = A276427(n,0).
G.f.: g(x) = Product_{i>=1} (1/(1-x^i) - x^{i^2}).
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EXAMPLE
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a(4) = 3 because we have [1,1,1,1], [1,1,2], and [4]; the partitions [1,3], [2,2] do not qualify.
From Gus Wiseman, Apr 02 2019: (Start)
The a(2) = 2 through a(7) = 9 partitions:
(2) (3) (4) (5) (6) (7)
(11) (111) (211) (32) (33) (43)
(1111) (311) (42) (52)
(2111) (222) (511)
(11111) (411) (3211)
(3111) (4111)
(21111) (31111)
(111111) (211111)
(1111111)
(End)
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MAPLE
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g := product(1/(1-x^i)-x^(i^2), i = 1 .. 100): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 0 .. 50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(`if`(i=j, 0, b(n-i*j, i-1)), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..60); # Alois P. Heinz, Sep 19 2016
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MATHEMATICA
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b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[If[i == j, x, 1]*b[n - i*j, i - 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n][[1]], {n, 0, 60}] (* Jean-François Alcover, Nov 28 2016 after Alois P. Heinz's Maple code for A276427 *)
Table[Length[Select[IntegerPartitions[n], And@@Table[Count[#, i]!=i, {i, Union[#]}]&]], {n, 0, 30}] (* Gus Wiseman, Apr 02 2019 *)
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CROSSREFS
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Cf. A052335, A087153, A114639, A115584, A117144, A276427, A324572, A325130, A325131, A336269.
Sequence in context: A114990 A241421 A157176 * A111181 A267419 A076777
Adjacent sequences: A276426 A276427 A276428 * A276430 A276431 A276432
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch, Sep 19 2016
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STATUS
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approved
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