%I #42 Aug 09 2024 05:03:43
%S 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,
%T 3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,
%U 5,5,5,5,5,5,5,5,5,5,5,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9,9,9
%N Number of partitions of n into fourth powers.
%C In general, the number of partitions of n into perfect s-th powers (s>=1) is asymptotic to (2*Pi)^(-(s+1)/2) * sqrt(s/(s+1)) * k * n^(1/(s+1)-3/2) * exp((s+1)*k*n^(1/(s+1))), where k = (Gamma(1 + 1/s) * Zeta(1 + 1/s) / s)^(s/(s+1)) [Hardy & Ramanujan, 1917]. - _Vaclav Kotesovec_, Dec 29 2016
%D H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
%H Alois P. Heinz, <a href="/A046042/b046042.txt">Table of n, a(n) for n = 1..10000</a>
%H G. H. Hardy and S. Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram33.html">Asymptotic formulae in combinatory analysis</a>, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.
%H Herman P. Robinson, <a href="/A003105/a003105.pdf">Letter to N. J. A. Sloane, Jan 1974</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Partition.html">Partition</a>
%F G.f.: -1+1/product(1-x^(j^4),j=1..infinity). - _Emeric Deutsch_, Apr 06 2006
%F a(n) ~ exp(5 * (Gamma(1/4)*Zeta(5/4))^(4/5) * n^(1/5) / 2^(16/5)) * (Gamma(1/4)*Zeta(5/4))^(4/5) / (2^(47/10) * sqrt(5) * Pi^(5/2) * n^(13/10)) [Hardy & Ramanujan, 1917]. - _Vaclav Kotesovec_, Dec 29 2016
%F G.f.: Sum_{i>=1} x^(i^4) / Product_{j=1..i} (1 - x^(j^4)). - _Ilya Gutkovskiy_, May 07 2017
%e a(33) = 3 because we have [16,16,1], [16,1,1,...,1] (17 1's) and [1,1,...,1] (33 1's).
%p g:=-1+1/product(1-x^(j^4),j=1..10): gser:=series(g,x=0,105): seq(coeff(gser,x,n),n=1..102); # _Emeric Deutsch_, Apr 06 2006
%t g = -1 + 1/Product[1 - x^(j^4), {j, 1, 10}]; gser =
%t Series[g, {x, 0, 105}]; Table[Coefficient[gser, x, n], {n, 1, 102}] (* _Jean-François Alcover_, Oct 29 2012, after _Emeric Deutsch_ *)
%o (Haskell)
%o a046042 = p $ tail a000583_list where
%o p _ 0 = 1
%o p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
%o -- _Reinhard Zumkeller_, May 18 2015 ~
%Y Cf. A000583, A002377, A003105.
%Y Cf. A001156, A003108, A046042.
%Y Cf. A037444, A259792, A259793.
%K nonn
%O 1,16
%A _Eric W. Weisstein_