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%I #7 Mar 10 2014 09:36:58
%S 0,0,0,0,1,1,4,4,9,9,18,18,31,31,51,51,79,79,119,119,173,173,248,248,
%T 347,347,480,480,654,654,883,883,1178,1178,1561,1561,2049,2049,2674,
%U 2674,3464,3464,4464,4464,5717,5717,7290,7290,9246,9246,11680,11680
%N Number of partitions p of n such that n - 2*(number of parts of p) is a part of p.
%H Giovanni Resta, <a href="/A238629/b238629.txt">Table of n, a(n) for n = 1..1000</a>
%e a(7) counts these partitions: 511, 43, 421, 331.
%t Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, n - 2*Length[p]]], {n, 50}]
%t p[n_, k_] := p[n, k] = If[k == 1 || n == k, 1, If[k > n, 0, p[n - 1, k - 1] + p[n - k, k]]]; q[n_, k_, e_] := q[n, k, e] = If[n - e < k - 1 , 0, If[k == 1, If[n == e, 1, 0], p[n - e, k - 1]]]; a[n_] := a[n] = Sum[q[n, u, n - 2*u], {u, (n - 1)/2}]; Array[a, 100] (* _Giovanni Resta_, Mar 09 2014 *)
%Y Cf. A000027 = (number of partitions p of n such that n - (number of parts of p) is a part of p) = n-2 for n >=3.
%K nonn,easy
%O 1,7
%A _Clark Kimberling_, Mar 02 2014