A218495 Number of partitions of n^2 into positive cubes.
1, 1, 1, 2, 3, 4, 7, 10, 17, 26, 39, 58, 89, 133, 195, 289, 420, 610, 875, 1253, 1778, 2514, 3527, 4937, 6879, 9516, 13115, 18012, 24625, 33503, 45432, 61402, 82677, 110913, 148286, 197722, 262768, 348100, 459791, 605780, 795874, 1042791, 1362800, 1776777
Offset: 0
Keywords
Examples
n=5: number of partitions of 25 into parts of {1, 8}:
a(5) = #{8+8+8+1, 8+8+9x1, 8+17x1, 25x1} = 4;
n=6: number of partitions of 36 into parts of {1, 8, 27}:
a(6) = #{27+8+1, 27+9x1, 4x8+4x1, 3x8+12x1, 8+8+20x1, 8+28x1, 36x1} = 7;
n=7: number of partitions of 49 into parts of {1, 8, 27}:
a(7) = #{27+8+8+6x1, 27+8+14x1, 27+22x1, 6x8+1, 5x8+9x1, 4x8+17x1, 3x8+25x1, 8+8+33x1, 8+41x1, 49x1} = 10.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
Programs
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Haskell
a218495 = p (tail a000578_list) . (^ 2) where p _ 0 = 1 p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
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Maple
b:= proc(n, i) option remember; `if`(n=0 or i=1, 1, b(n, i-1) +`if`(i^3>n, 0, b(n-i^3, i))) end: a:= n-> b(n^2, iroot(n^2, 3)): seq(a(n), n=0..60); # Alois P. Heinz, Nov 08 2012 -
Mathematica
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i-1] + If[i^3>n, 0, b[n - i^3, i]]]; a[n_] := b[n^2, n^(2/3) // Floor]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *) -
PARI
a(n) = {my(nb=0); forpart(p=n^2, nb += (sum(k=1, #p, ispower(p[k], 3)) == #p);); nb;} \\ Michel Marcus, Jun 02 2015 -
PARI
ok(p)=for(i=1,#p,if(!ispower(p[i],3),return(0)));1 a(n)=my(s=1);for(i=8,n^2,forpart(p=i,s+=ok(p),[8,sqrtnint(i,3)^3]));s \\ Charles R Greathouse IV, Jun 02 2015
Formula
a(n) ~ exp(4 * (Gamma(1/3)*Zeta(4/3))^(3/4) * sqrt(n) / 3^(3/2)) * (Gamma(1/3)*Zeta(4/3))^(3/4) / (24*Pi^2*n^(5/2)) [after Hardy & Ramanujan, 1917]. - Vaclav Kotesovec, Apr 10 2017
a(n) = [x^(n^2)] Product_{k>=1} 1/(1 - x^(k^3)). - Ilya Gutkovskiy, Jun 05 2017
Extensions
More terms from Alois P. Heinz, Nov 08 2012
Comments