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A218705
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Number of partitions of n in which any two distinct parts differ by at least 10.
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2
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1, 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 7, 4, 8, 9, 13, 11, 18, 16, 23, 22, 27, 27, 36, 35, 41, 42, 51, 48, 61, 57, 69, 65, 80, 81, 98, 93, 115, 112, 144, 136, 170, 164, 202, 204, 244, 242, 296, 290, 353, 350, 415, 412, 493, 494, 576, 580, 671, 673, 794, 786, 903
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OFFSET
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0,3
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COMMENTS
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Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 10 times.
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LINKS
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FORMULA
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G.f.: 1 + Sum_{j>=1} x^j/(1-x^j) * Product_{i=1..j-1} (1+x^(10*i)/(1-x^i)).
log(a(n)) ~ sqrt((2*Pi^2/3 + 4*c)*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-10*x)) dx = -1.2055372531240537414216314471404302128615809819... - Vaclav Kotesovec, Jan 28 2022
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EXAMPLE
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a(10) = 4: [1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [10].
a(11) = 2: [1,1,1,1,1,1,1,1,1,1,1], [11].
a(12) = 7: [1,1,1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2,2], [3,3,3,3], [4,4,4], [6,6], [1,11], [12].
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1) +add(b(n-i*j, i-10), j=1..n/i)))
end:
a:= n-> b(n, n):
seq(a(n), n=0..70);
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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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