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A089333
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Number of partitions into a square number of parts.
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7
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1, 1, 1, 1, 2, 2, 3, 4, 6, 8, 11, 14, 19, 24, 31, 39, 51, 63, 80, 99, 124, 153, 190, 233, 288, 353, 432, 527, 643, 780, 947, 1145, 1383, 1665, 2002, 2399, 2874, 3431, 4090, 4865, 5779, 6847, 8103, 9568, 11283, 13280, 15610, 18313, 21462, 25108, 29337, 34227
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OFFSET
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0,5
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COMMENTS
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Also number of partitions of n such that the largest part is a square. Example: a(7)=4 because we have [4,3], [4,2,1], [4,1,1,1] and [1,1,1,1,1,1,1]. - Emeric Deutsch, Apr 04 2006
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..10000
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FORMULA
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G.f.: Sum(x^(n^2)/Product(1-x^i, i = 1 .. n^2), n = 1 .. infinity).
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EXAMPLE
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a(7)=4 because we have [7], [4,1,1,1], [3,2,1,1] and [2,2,2,1].
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MAPLE
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g:=sum(x^(k^2)/product(1-x^i, i=1..k^2), k=1..7): gser:=series(g, x=0, 55): seq(coeff(gser, x, n), n=1..51); # Emeric Deutsch, Apr 04 2006
# second Maple program:
b:= proc(n, i) option remember; `if`(n<0, 0,
`if`(n=0 or i=1, 1, `if`(i<1, 0, b(n, i-1)+
`if`(i>n, 0, b(n-i, i)))))
end:
a:= n-> add(b(n-i^2, i^2), i=0..isqrt(n)):
seq(a(n), n=0..60); # Alois P. Heinz, Sep 24 2015
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n < 0, 0, If[n == 0 || i == 1, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i]]]]]; a[n_] := Sum[b[n - i^2, i^2], {i, 0, Sqrt[n]}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Jan 10 2016, after Alois P. Heinz*)
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CROSSREFS
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Sequence in context: A130081 A141847 A182372 * A098492 A217314 A173508
Adjacent sequences: A089330 A089331 A089332 * A089334 A089335 A089336
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KEYWORD
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easy,nonn
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AUTHOR
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Vladeta Jovovic, Dec 25 2003
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EXTENSIONS
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a(0)=1 from Alois P. Heinz, Sep 24 2015
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STATUS
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approved
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