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A363214
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Number of partitions of n with rank 5 (the rank of a partition is the largest part minus the number of parts).
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2
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0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 7, 11, 13, 19, 21, 30, 35, 47, 55, 73, 85, 111, 131, 166, 197, 247, 291, 362, 428, 525, 620, 756, 890, 1078, 1268, 1523, 1791, 2140, 2507, 2983, 3490, 4131, 4824, 5688, 6626, 7785, 9052, 10595, 12298, 14351, 16618, 19339, 22355, 25938
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OFFSET
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1,10
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LINKS
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FORMULA
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G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(5*k) * ( x^(k*(3*k-1)/2) - x^(k*(3*k+1)/2) ).
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MATHEMATICA
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Table[Count[IntegerPartitions[n], _?(#[[1]]-Length[#]==5&)], {n, 60}] (* Harvey P. Dale, Feb 24 2024 *)
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PROG
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(PARI) my(N=60, x='x+O('x^N)); concat([0, 0, 0, 0, 0], Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(5*k)*(x^(k*(3*k-1)/2)-x^(k*(3*k+1)/2)))))
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CROSSREFS
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Column k=5 in the triangle A063995.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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