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A000715
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Number of partitions of n, with three kinds of 1,2 and 3 and two kinds of 4,5,6,....
(Formerly M2786 N1121)
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2
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1, 3, 9, 22, 50, 104, 208, 394, 724, 1286, 2229, 3769, 6253, 10176, 16303, 25723, 40055, 61588, 93647, 140875, 209889, 309846, 453565, 658627, 949310, 1358589, 1931464, 2728547, 3831654, 5350119, 7430158, 10265669, 14113795, 19313168, 26309405, 35685523
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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, 2, 2, 2, 2, 2, ...
G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*Product_{k>=1}(1-x^k)^2). - Emeric Deutsch, Apr 17 2006
a(n) ~ exp(2*Pi*sqrt(n/3)) * n^(1/4) / (8 * 3^(1/4) * Pi^3). - 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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g:=1/((1-x)*(1-x^2)*(1-x^3)*product((1-x^k)^2, k=1..40)): gser:=series(g, x=0, 40): seq(coeff(gser, x, n), n=0..31); # Emeric Deutsch, Apr 17 2006
# second Maple program
a:= proc(n) a(n):= `if`(n=0, 1, add(add(d*`if`(d<4, 3, 2), d=numtheory [divisors](j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..50); # Alois P. Heinz, Sep 25 2012
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MATHEMATICA
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nn=25; p=Product[1/(1- x^i)^2, {i, 1, nn}]; CoefficientList[Series[p /(1-x)/(1-x^2)/(1-x^3), {x, 0, nn}], x] (* Geoffrey Critzer, Sep 25 2012 *)
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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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