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A363231
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Number of partitions of n with rank 4 or higher (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, 1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, 53, 72, 94, 124, 161, 209, 267, 343, 435, 551, 693, 870, 1084, 1351, 1672, 2066, 2542, 3121, 3815, 4658, 5664, 6875, 8319, 10049, 12102, 14553, 17452, 20894, 24959, 29766, 35420, 42089, 49911, 59100, 69856, 82452, 97152, 114324, 134315
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OFFSET
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1,7
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COMMENTS
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In general, for r>=0, Sum_{k>=1} (-1)^(k-1) * p(n - k*(3*k + 2*r - 1)/2) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)) * (1 - (1/(2*Pi) + (12*r-5)*Pi/144) / sqrt(n/6)), where p() is the partition function. - Vaclav Kotesovec, May 26 2023
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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^(k*(3*k+7)/2).
a(n) = p(n-5) - p(n-13) + p(n-24) - ... + (-1)^(k-1) * p(n-k*(3*k+7)/2) + ..., where p() is A000041().
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)) * (1 - (1/(2*Pi) + 43*Pi/144) / sqrt(n/6)). - Vaclav Kotesovec, May 26 2023
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EXAMPLE
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a(7) = 2 counts these partitions: 7, 6+1.
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MATHEMATICA
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Table[Count[IntegerPartitions[n], _?(#[[1]]-Length[#]>3&)], {n, 60}] (* Harvey P. Dale, Jul 29 2024 *)
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PROG
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(PARI) a(n) = sum(k=1, sqrtint(n), (-1)^(k-1)*numbpart(n-k*(3*k+7)/2));
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CROSSREFS
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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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