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A320689
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Number of partitions of n with up to two distinct kinds of 1.
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2
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1, 2, 2, 3, 5, 7, 10, 14, 19, 26, 35, 46, 61, 80, 103, 133, 171, 217, 275, 347, 435, 544, 677, 838, 1036, 1276, 1564, 1913, 2334, 2837, 3441, 4163, 5022, 6046, 7262, 8701, 10407, 12421, 14792, 17586, 20871, 24721, 29234, 34514, 40679, 47874, 56256, 66003
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * sqrt(2) * n^(3/2)). - Vaclav Kotesovec, Oct 24 2018
G.f.: (1 + x)^2 * Product_{k>=2} 1 / (1 - x^k). - Ilya Gutkovskiy, Apr 24 2021
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0 or i=1,
binomial(2, n), `if`(i>n, 0, b(n-i, i))+b(n, i-1))
end:
a:= n-> b(n$2):
seq(a(n), n=0..60);
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n == 0 || i == 1, Binomial[2, n], If[i > n, 0, b[n - i, i]] + b[n, i - 1]];
a[n_] := b[n, n];
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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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