OFFSET
0,3
COMMENTS
First term a(0) may be computed as 1 by starting the inner sum at j=0 and taking the convention 0^0 = 1.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
a(n) = Sum_{k=1..n} Sum_{j=1..k} j^n.
a(n) = Sum_{k=1..n} H_k^{-n} where H_k^{-n} is the k-th harmonic number of order -n.
a(n) = Sum_{k=1..n} (B(n+1, k+1) - B(n+1, 1))/(n+1), where B(n, x) are the Bernoulli polynomials. - Daniel Suteu, Jun 25 2018
G.f.: Sum_{k>=1} k^k*x^k/(1 - k*x)^2. - Ilya Gutkovskiy, Oct 11 2018
a(n) ~ c * n^n, where c = 1/(1 - 2*exp(-1) + exp(-2)) = 2.50265030107711874333... - Vaclav Kotesovec, Nov 06 2021
EXAMPLE
a(3) = (1^3) + (1^3 + 2^3) + (1^3 + 2^3 + 3^3) = (1^3 + 1^3 + 1^3) + (2^3 + 2^3) + (3^3) = 3 * 1^3 + 2 * 2^3 + 1 * 3^3 = 46. - David A. Corneth, Jun 27 2018
MATHEMATICA
Table[Sum[Sum[j^n, {j, 1, k}], {k, 0, n}], {n, 0, 20}]
a[n_] := (n+1)*HarmonicNumber[-1, -n] - HarmonicNumber[n, -n-1] + (n+1)*HarmonicNumber[n, -n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 05 2013 *)
Table[Total[Accumulate[Range[n]^n]], {n, 0, 20}] (* Harvey P. Dale, Mar 29 2020 *)
PROG
(PARI) a(n) = sum(k=1, n, sum(j=1, k, j^n)); \\ Michel Marcus, Jun 25 2018
(PARI) a(n) = sum(i=1, n, (n+1-i) * i^n); \\ David A. Corneth, Jun 27 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Olivier Gérard, Aug 02 2012
STATUS
approved