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 A037444 Number of partitions of n^2 into squares. 39
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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 LINKS 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. FORMULA 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 MAPLE 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 MATHEMATICA 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 *) PROG (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 CROSSREFS 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 KEYWORD nonn,nice,easy AUTHOR STATUS approved

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Last modified April 2 19:05 EDT 2020. Contains 333190 sequences. (Running on oeis4.)