OFFSET
0,3
COMMENTS
a(0) could alternatively be defined as 1 from the formula or the convention for 0^0.
This sum is remarkable for its three different decompositions involving powers and binomials (see formulas and cross-refs).
LINKS
Winston de Greef, Table of n, a(n) for n = 0..372 (first 201 terms from Vincenzo Librandi)
FORMULA
a(n) = Sum_{k=0..n} binomial(n,k)*Sum_{j=1..k} j^n;
a(n) = Sum_{k=0..n} binomial(n,k)*H_k^{-n}, where H_k^(-n) = k-th harmonic number of order -n;
a(n) = Sum_{k=0..n} k^n * Sum_{j=0..n-k} binomial(n,n-k-j);
a(n) = Sum_{k=0..n} k^n * binomial(n,n-k) * 2F1(1, k-n; k+1)(-1);
a(n) = Sum_{k=0..n} Sum_{j=0..k} (k-j)^n * binomial(n,j);
a(n) = Sum_{k=0..n} Sum_{j=0..n} (n-j)^n * binomial(n,n+k-j);
and the equivalent formulas obtained by symmetries of the binomial and the hypergeometric function as well as treating the zeroth term separately.
a(n) ~ n^n / (sqrt(1+r) * (1-r) * exp(n) * r^n), where r = A202357 = LambertW(exp(-1)). - Vaclav Kotesovec, Jun 10 2019
MATHEMATICA
Table[Sum[Sum[j^n*Binomial[n, k], {j, 1, k}], {k, 0, n}], {n, 0, 20}]
PROG
(PARI) a(n)=sum(k=0, n, binomial(n, k)*sum(j=1, k, j^n)) \\ Charles R Greathouse IV, Jul 31 2016
(PARI) a(n)=my(P=sumformal('x^n)); sum(k=0, n, binomial(n, k)*subst(P, 'x, k)) \\ Charles R Greathouse IV, Jul 31 2016
KEYWORD
nonn,nice
AUTHOR
Olivier Gérard, Aug 02 2012
STATUS
approved