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A237755
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Number of partitions of n such that 2*(greatest part) >= (number of parts).
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24
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1, 2, 2, 4, 6, 9, 12, 18, 24, 34, 46, 63, 83, 111, 144, 190, 245, 318, 405, 520, 657, 833, 1045, 1312, 1634, 2036, 2517, 3114, 3829, 4705, 5751, 7027, 8544, 10381, 12564, 15190, 18301, 22026, 26425, 31669, 37849, 45180, 53796, 63983, 75923, 89987, 106435
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OFFSET
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1,2
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COMMENTS
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Also, the number of partitions of n such that (greatest part) <= 2*(number of parts); hence, the number of partitions of n such that (rank + greatest part) >= 0.
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LINKS
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FORMULA
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G.f.: Sum_{n>=1} x^n * Product_{k=1..n} (1 - x^(2*n+k-1))/(1 - x^k). - Paul D. Hanna, Aug 03 2015
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EXAMPLE
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a(6) = 9 counts all of the 11 partitions of 6 except these: 21111, 111111.
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MATHEMATICA
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z = 50; Table[Count[IntegerPartitions[n], p_ /; 2 Max[p] >= Length[p]], {n, z}]
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PROG
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(PARI) {a(n) = my(A); A = sum(m=0, n, x^m*prod(k=1, m, (1-x^(2*m+k-1))/(1-x^k +x*O(x^n)))); polcoeff(A, n)}
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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