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A000711
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Number of partitions of n, with three kinds of 1,2,3 and 4 and two kinds of 5,6,7,...
(Formerly M2787 N1122)
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9
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1, 3, 9, 22, 51, 107, 217, 416, 775, 1393, 2446, 4185, 7028, 11569, 18749, 29908, 47083, 73157, 112396, 170783, 256972, 383003, 565961, 829410, 1206282, 1741592, 2497425, 3557957, 5037936, 7091711, 9927583, 13823626, 19151731, 26404879, 36236988, 49509149
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OFFSET
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0,2
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COMMENTS
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REFERENCES
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H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 122.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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EULER transform of 3, 3, 3, 3, 2, 2, 2, 2, ...
G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*Product_{k>=1} (1 - x^k)^2).
a(n) ~ exp(2*Pi*sqrt(n/3)) * 3^(1/4) * n^(3/4) / (32*Pi^4). - Vaclav Kotesovec, Aug 18 2015
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EXAMPLE
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a(2)=9 because we have 2, 2', 2", 1+1, 1'+1', 1"+1", 1+1', 1+1", 1'+1".
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MAPLE
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with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add (add (d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> `if`(n<5, 3, 2)): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008
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MATHEMATICA
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nn=31; CoefficientList[Series[1/(1-x)/(1-x^2)/(1-x^3)/(1-x^4)/Product[(1-x^i)^2, {i, 1, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Sep 28 2013 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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