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%I #9 Apr 24 2014 10:27:29
%S 1,1,1,2,4,4,8,8,12,16,23,24,40,45,59,72,99,108,153,171,224,263,341,
%T 377,504,567,711,821,1035,1153,1467,1648,2028,2317,2841,3171,3923,
%U 4403,5308,6014,7250,8095,9778,10949,13018,14672,17400,19405,23061,25769,30243
%N Number of partitions of n having exactly 1 part that appears exactly once.
%C Column 1 of A116595.
%F G.f.=sum(x^j*(1-x^j)/(1-x^j+x^(2j)), j=1..infinity)product((1-x^j+x^(2j))/(1-x^j), j=1..infinity).
%F G.f. for number of partitions of n having exactly 1 part that appears exactly m times is sum(x^(m*j)*(1-x^j)/(1-x^(m*j)+x^((m+1)*j)), j=1..infinity)*product((1-x^(m*j)+x^((m+1)*j))/(1-x^j), j=1..infinity). - _Vladeta Jovovic_, May 01 2006
%e a(5)=4 because we have [5],[3,1,1],[2,2,1] and [2,1,1,1] ([4,1],[3,2] and [1,1,1,1,1] do not qualify).
%p f:=sum(x^j*(1-x^j)/(1-x^j+x^(2*j)),j=1..75)*product((1-x^j+x^(2*j))/(1-x^j),j=1..75): fser:=series(f,x=0,73): seq(coeff(fser,x^n),n=1..55);
%t z = 30; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]]; m1[p_] := Min[Map[Length, Split[p]]]; Table[Count[IntegerPartitions[n], p_ /; u[p] == m1[p]], {n, 0, z}] (* _Clark Kimberling_, Apr 23 2014 *)
%Y Cf. A116595.
%K nonn
%O 1,4
%A _Emeric Deutsch_, Feb 18 2006
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