A259793 Number of partitions of n^4 into fourth powers.
1, 1, 2, 7, 36, 253, 1886, 14800, 118238, 955639, 7750456, 62777522, 506272363, 4056634991, 32252971687, 254209569990, 1985108901344, 15352968310930, 117579612410477, 891596419221856, 6694250497509934, 49768995849050468, 366423320400440927, 2671969175372760210
Offset: 0
Keywords
Links
- Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..63 (terms 0..45 from Alois P. Heinz)
- 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.
Programs
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Maple
b:= proc(n, i) option remember; `if`(n=0 or i=1, 1, b(n, i-1) +`if`(i^4>n, 0, b(n-i^4, i))) end: a:= n-> b(n^4, n): seq(a(n), n=0..23); # Alois P. Heinz, Jul 10 2015 -
Mathematica
$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 *)
Formula
a(n) = [x^(n^4)] Product_{j>=1} 1/(1-x^(j^4)). - Alois P. Heinz, Jul 10 2015
a(n) = A046042(n^4). - Vaclav Kotesovec, Aug 19 2015
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
Extensions
More terms from Alois P. Heinz, Jul 10 2015