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Number of partitions of n into fourth powers.
15

%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_