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Giovanni Resta, <a href="/A238629/b238629.txt">Table of n, a(n) for n = 1..1000</a>
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 *)
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allocated for Clark KimberlingNumber of partitions p of n such that n - 2*(number of parts of p) is a part of p.
0, 0, 0, 0, 1, 1, 4, 4, 9, 9, 18, 18, 31, 31, 51, 51, 79, 79, 119, 119, 173, 173, 248, 248, 347, 347, 480, 480, 654, 654, 883, 883, 1178, 1178, 1561, 1561, 2049, 2049, 2674, 2674, 3464, 3464, 4464, 4464, 5717, 5717, 7290, 7290, 9246, 9246, 11680, 11680
1,7
a(7) counts these partitions: 511, 43, 421, 331.
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, n - 2*Length[p]]], {n, 50}]
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.
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Clark Kimberling, Mar 02 2014
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allocated for Clark Kimberling
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