|
|
A229325
|
|
Total sum of cubes of parts in all partitions of n.
|
|
3
|
|
|
0, 1, 10, 39, 122, 287, 660, 1281, 2486, 4392, 7686, 12628, 20790, 32471, 50694, 76560, 115038, 168333, 245784, 350896, 499620, 699468, 975150, 1341077, 1838550, 2490092, 3361260, 4494084, 5986750, 7909231, 10416300, 13616768, 17745948, 22983345, 29672974
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
The bivariate g.f. for the partition statistic "sum of cubes of the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^3}*x^k). The g.f. g given in the Formula section was obtained by evaluating dG/dt at t=1. - Emeric Deutsch, Dec 06 2015
|
|
LINKS
|
|
|
FORMULA
|
a(n) = Sum_{k=1..n} A066633(n,k) * k^3.
G.f.: g(x) = (Sum_{k>=1} k^3*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - Emeric Deutsch, Dec 06 2015
|
|
MAPLE
|
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
`if`(i<1, [0, 0], `if`(i>n, b(n, i-1),
((g, h)-> g+h+[0, h[1]*i^3])(b(n, i-1), b(n-i, i)))))
end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..40);
|
|
MATHEMATICA
|
Table[Total[Flatten[IntegerPartitions[n]^3]], {n, 0, 40}] (* Harvey P. Dale, May 01 2016 *)
b[n_, i_] := b[n, i] = If[n==0, {1, 0}, If[i<1, {0, 0}, If[i>n, b[n, i-1], Function[{g, h}, g + h + {0, h[[1]]*i^3}][b[n, i-1], b[n-i, i]]]]];
a[n_] := b[n, n][[2]];
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|