A215077 Binomial convolution of sum of consecutive powers.

0, 1, 7, 66, 852, 14020, 280472, 6609232, 179317056, 5505532992, 188717617280, 7143999854464, 296013377405440, 13325516967972352, 647610246703508480, 33794224057227356160, 1884620857353101983744, 111857608180484932648960, 7040178644779119413723136, 468349192560992552808841216, 32836927387372039917034405888

Offset: 0

Author

Olivier Gérard, Aug 02 2012

Keywords

• nonn
• nice

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)

Crossrefs

Row sums of A215078, A215079, A215080.

See also A215083 and A215084.

Programs

Mathematica

Table[Sum[Sum[j^n*Binomial[n, k], {j, 1, k}], {k, 0, n}], {n, 0, 20}]

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

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