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A259793 Number of partitions of n^4 into fourth powers.

%I A259793 #23 Dec 29 2016 16:02:10
%S A259793 1,1,2,7,36,253,1886,14800,118238,955639,7750456,62777522,506272363,
%T A259793 4056634991,32252971687,254209569990,1985108901344,15352968310930,
%U A259793 117579612410477,891596419221856,6694250497509934,49768995849050468,366423320400440927,2671969175372760210
%N A259793 Number of partitions of n^4 into fourth powers.
%H A259793 Alois P. Heinz and Vaclav Kotesovec, <a href="/A259793/b259793.txt">Table of n, a(n) for n = 0..63</a> (terms 0..45 from Alois P. Heinz)
%H A259793 H. L. Fisher, <a href="/A027601/a027601.pdf">Letter to N. J. A. Sloane, Mar 16 1989</a>
%H A259793 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.
%F A259793 a(n) = [x^(n^4)] Product_{j>=1} 1/(1-x^(j^4)). - _Alois P. Heinz_, Jul 10 2015
%F A259793 a(n) = A046042(n^4). - _Vaclav Kotesovec_, Aug 19 2015
%F A259793 a(n) ~ exp(5 * (Gamma(1/4)*Zeta(5/4))^(4/5) * n^(4/5) / 2^(16/5)) * (Gamma(1/4)*Zeta(5/4))^(4/5) / (2^(47/10) * sqrt(5) * Pi^(5/2) * n^(26/5)) [after Hardy & Ramanujan, 1917]. - _Vaclav Kotesovec_, Dec 29 2016
%p A259793 b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p A259793       b(n, i-1) +`if`(i^4>n, 0, b(n-i^4, i)))
%p A259793     end:
%p A259793 a:= n-> b(n^4, n):
%p A259793 seq(a(n), n=0..23);  # _Alois P. Heinz_, Jul 10 2015
%t A259793 $RecursionLimit = 10^4; b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1] + If[i^4>n, 0, b[n-i^4, i]]]; a[n_] := b[n^4, n];  Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Dec 06 2016 after _Alois P. Heinz_ *)
%Y A259793 A row of the array in A259799.
%Y A259793 Cf. A001156, A003108, A046042.
%Y A259793 Cf. A037444, A259792.
%K A259793 nonn
%O A259793 0,3
%A A259793 _N. J. A. Sloane_, Jul 06 2015
%E A259793 More terms from _Alois P. Heinz_, Jul 10 2015