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A117144 Partitions of n in which each part k occurs at least k times.

%I A117144 #21 Jan 28 2024 09:31:23
%S A117144 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,
%T A117144 55,64,70,79,86,97,106,119,129,146,159,175,190,214,232,256,277,306,
%U A117144 334,367,394,434,472,515,556,607,654,714,770,836,901,978,1048,1140,1226,1322
%N A117144 Partitions of n in which each part k occurs at least k times.
%C A117144 The Heinz numbers of these integer partitions are given by A324525. - _Gus Wiseman_, Mar 09 2019
%H A117144 Alois P. Heinz, <a href="/A117144/b117144.txt">Table of n, a(n) for n = 0..10000</a>
%F A117144 G.f.: Product_{k>=1} (1-x^k+x^(k^2))/(1-x^k).
%e A117144 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].
%e A117144 From _Gus Wiseman_, Mar 09 2019: (Start)
%e A117144 The a(1) = 1 through a(9) = 5 integer partitions:
%e A117144   1  11  111  22    221    222     2221     2222      333
%e A117144               1111  11111  2211    22111    22211     22221
%e A117144                            111111  1111111  221111    222111
%e A117144                                             11111111  2211111
%e A117144                                                       111111111
%e A117144 (End)
%p A117144 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);
%p A117144 # second Maple program:
%p A117144 b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A117144       b(n, i-1) +add(b(n-i*j, i-1), j=i..n/i)))
%p A117144     end:
%p A117144 a:= n-> b(n$2):
%p A117144 seq(a(n), n=0..80);  # _Alois P. Heinz_, Dec 28 2016
%t A117144 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_ *)
%t A117144 Table[Length[Select[IntegerPartitions[n],And@@Table[Count[#,i]>=i,{i,Union[#]}]&]],{n,0,30}] (* _Gus Wiseman_, Mar 09 2019 *)
%t A117144 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 *)
%Y A117144 Cf. A001462, A003114, A006141, A033461, A039900, A047993, A052335, A064174, A090858, A114638, A115584, A276078, A280204.
%Y A117144 Cf. A324518, A324520, A324525, A324572.
%K A117144 nonn
%O A117144 0,5
%A A117144 _Emeric Deutsch_, Mar 06 2006