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

0,3

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

Crossrefs

Row sums of A215078, A215079, A215080.

See also A215083 and A215084.

Sequence in context: A122705 A185181 A024395 * A003286 A244602 A223889

Adjacent sequences: A215074 A215075 A215076 * A215078 A215079 A215080

Keyword

nonn,nice

Author

Olivier Gérard, Aug 02 2012

Status

approved