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A276429 Number of partitions of n containing no part i of multiplicity i.

%I A276429 #25 Jul 15 2020 15:15:10
%S A276429 1,0,2,2,3,5,8,9,16,19,29,36,53,65,92,115,154,195,257,318,419,516,663,
%T A276429 821,1039,1277,1606,1963,2441,2978,3675,4454,5469,6603,8043,9688,
%U A276429 11732,14066,16963,20260,24310,28953,34586,41047,48857,57802,68528,80862,95534,112388,132391
%N A276429 Number of partitions of n containing no part i of multiplicity i.
%C A276429 The Heinz numbers of these partitions are given by A325130. - _Gus Wiseman_, Apr 02 2019
%H A276429 Vaclav Kotesovec, <a href="/A276429/b276429.txt">Table of n, a(n) for n = 0..12782</a> (terms 0..5000 from Alois P. Heinz)
%F A276429 a(n) = A276427(n,0).
%F A276429 G.f.: g(x) = Product_{i>=1} (1/(1-x^i) - x^{i^2}).
%e A276429 a(4) = 3 because we have [1,1,1,1], [1,1,2], and [4]; the partitions [1,3], [2,2] do not qualify.
%e A276429 From _Gus Wiseman_, Apr 02 2019: (Start)
%e A276429 The a(2) = 2 through a(7) = 9 partitions:
%e A276429   (2)   (3)    (4)     (5)      (6)       (7)
%e A276429   (11)  (111)  (211)   (32)     (33)      (43)
%e A276429                (1111)  (311)    (42)      (52)
%e A276429                        (2111)   (222)     (511)
%e A276429                        (11111)  (411)     (3211)
%e A276429                                 (3111)    (4111)
%e A276429                                 (21111)   (31111)
%e A276429                                 (111111)  (211111)
%e A276429                                           (1111111)
%e A276429 (End)
%p A276429 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);
%p A276429 # second Maple program:
%p A276429 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A276429       add(`if`(i=j, 0, b(n-i*j, i-1)), j=0..n/i)))
%p A276429     end:
%p A276429 a:= n-> b(n$2):
%p A276429 seq(a(n), n=0..60);  # _Alois P. Heinz_, Sep 19 2016
%t A276429 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 *)
%t A276429 Table[Length[Select[IntegerPartitions[n],And@@Table[Count[#,i]!=i,{i,Union[#]}]&]],{n,0,30}] (* _Gus Wiseman_, Apr 02 2019 *)
%Y A276429 Cf. A052335, A087153, A114639, A115584, A117144, A276427, A324572, A325130, A325131, A336269.
%K A276429 nonn
%O A276429 0,3
%A A276429 _Emeric Deutsch_, Sep 19 2016