|
|
A229326
|
|
Total sum of 4th powers of parts in all partitions of n.
|
|
2
|
|
|
0, 1, 18, 101, 392, 1119, 2904, 6407, 13578, 26218, 49218, 86782, 150860, 249723, 408810, 647170, 1013278, 1545029, 2337738, 3460218, 5086658, 7350874, 10549872, 14929931, 21009874, 29205500, 40385036, 55289000, 75309056, 101692923, 136710130, 182377824
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
The bivariate g.f. for the partition statistic "sum of 4th powers of the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^4}*x^k). The g.f. g at the Formula section has been 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^4.
G.f.: g(x) = (Sum_{k>=1} k^4*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - Emeric Deutsch, Dec 06 2015
a(n) ~ 216*sqrt(2)*Zeta(5)/Pi^5 * exp(Pi*sqrt(2*n/3)) * n^(3/2). - Vaclav Kotesovec, May 28 2018
|
|
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^4])(b(n, i-1), b(n-i, i)))))
end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..40);
# second Maple program:
g := (sum(k^4*x^k/(1-x^k), k = 1..100))/(product(1-x^k, k = 1..100)): gser := series(g, x = 0, 45): seq(coeff(gser, x, m), m = 1 .. 40); # Emeric Deutsch, Dec 06 2015
|
|
MATHEMATICA
|
(* T = A066633 *) T[n_, n_] = 1; T[n_, k_] /; k < n := T[n, k] = T[n-k, k] + PartitionsP[n-k]; T[_, _] = 0; a[n_] := Sum[T[n, k]*k^4, {k, 1, n}]; Array[a, 32, 0] (* Jean-François Alcover, Dec 15 2016 *)
Table[Sum[DivisorSigma[4, k]*PartitionsP[n-k], {k, 1, n}], {n, 0, 40}] (* Vaclav Kotesovec, May 27 2018 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|