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A037444
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Number of partitions of n^2 into squares.
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38
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1, 1, 2, 4, 8, 19, 43, 98, 220, 504, 1116, 2468, 5368, 11592, 24694, 52170, 108963, 225644, 462865, 941528, 1899244, 3801227, 7550473, 14889455, 29159061, 56722410, 109637563, 210605770, 402165159, 763549779, 1441686280, 2707535748, 5058654069, 9404116777
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OFFSET
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0,3
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COMMENTS
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Is lim_{n->inf} a(n)^(1/n) > 1? - Paul D. Hanna, Aug 20 2002
The limit above is equal to 1 (see formula by Hardy & Ramanujan for A001156). - Vaclav Kotesovec, Dec 29 2016
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LINKS
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T. D. Noe, Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..945 (terms n = 0..100 from T. D. Noe, terms n = 101..500 from Alois P. Heinz)
J. Bohman et al., Partitions in squares, Nordisk Tidskr. Informationsbehandling (BIT) 19 (1979), 297-301.
H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989
G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.
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FORMULA
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a(n) = A001156(n^2) = coefficient of x^(n^2) in the series expansion of Prod_{k>=1} 1/(1 - x^(k^2)).
a(n) ~ 3^(-1/2) * (4*Pi)^(-7/6) * Zeta(3/2)^(2/3) * n^(-7/3) * exp(2^(-4/3) * 3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(2/3)) [Hardy & Ramanujan, 1917, modified from A001156]. - Vaclav Kotesovec, Dec 29 2016
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
b(n, i-1) +`if`(i^2>n, 0, b(n-i^2, i)))
end:
a:= n-> b(n^2, n):
seq(a(n), n=0..40); # Alois P. Heinz, Apr 15 2013
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MATHEMATICA
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max=33; se = Series[ Product[1/(1-x^(k^2)), {k, 1, max}], {x, 0, max^2}]; a[n_] := Coefficient[se, x^(n^2)]; a[0] = 1; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Oct 18 2011 *)
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PROG
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(Haskell)
a037444 n = p (map (^ 2) [1..]) (n^2) where
p _ 0 = 1
p ks'@(k:ks) m | m < k = 0
| otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 14 2011
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CROSSREFS
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Entries with square index in A001156.
Cf. A072964, A030273, A000041, A000290, A229239, A229468.
Cf. A003108, A046042.
Cf. A259792, A259793.
A row or column of the array in A259799.
Sequence in context: A247235 A261663 A199694 * A151526 A099526 A005703
Adjacent sequences: A037441 A037442 A037443 * A037445 A037446 A037447
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KEYWORD
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nonn,nice,easy
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AUTHOR
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Wouter Meeussen
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STATUS
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approved
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