cp's OEIS Frontend

A215077 Binomial convolution of sum of consecutive powers.

%I A215077 #30 Jan 29 2023 20:56:03
%S A215077 0,1,7,66,852,14020,280472,6609232,179317056,5505532992,188717617280,
%T A215077 7143999854464,296013377405440,13325516967972352,647610246703508480,
%U A215077 33794224057227356160,1884620857353101983744,111857608180484932648960,7040178644779119413723136,468349192560992552808841216,32836927387372039917034405888
%N A215077 Binomial convolution of sum of consecutive powers.
%C A215077 a(0) could alternatively be defined as 1 from the formula or the convention for 0^0.
%C A215077 This sum is remarkable for its three different decompositions involving powers and binomials (see formulas and cross-refs).
%H A215077 Winston de Greef, <a href="/A215077/b215077.txt">Table of n, a(n) for n = 0..372</a> (first 201 terms from Vincenzo Librandi)
%F A215077 a(n) = Sum_{k=0..n} binomial(n,k)*Sum_{j=1..k} j^n;
%F A215077 a(n) = Sum_{k=0..n} binomial(n,k)*H_k^{-n}, where H_k^(-n) = k-th harmonic number of order -n;
%F A215077 a(n) = Sum_{k=0..n} k^n * Sum_{j=0..n-k} binomial(n,n-k-j);
%F A215077 a(n) = Sum_{k=0..n} k^n * binomial(n,n-k) * 2F1(1, k-n; k+1)(-1);
%F A215077 a(n) = Sum_{k=0..n} Sum_{j=0..k} (k-j)^n * binomial(n,j);
%F A215077 a(n) = Sum_{k=0..n} Sum_{j=0..n} (n-j)^n * binomial(n,n+k-j);
%F A215077 and the equivalent formulas obtained by symmetries of the binomial and the hypergeometric function as well as treating the zeroth term separately.
%F A215077 a(n) ~ n^n / (sqrt(1+r) * (1-r) * exp(n) * r^n), where r = A202357 = LambertW(exp(-1)). - _Vaclav Kotesovec_, Jun 10 2019
%t A215077 Table[Sum[Sum[j^n*Binomial[n, k], {j, 1, k}], {k, 0, n}], {n, 0, 20}]
%o A215077 (PARI) a(n)=sum(k=0,n, binomial(n,k)*sum(j=1,k, j^n)) \\ _Charles R Greathouse IV_, Jul 31 2016
%o A215077 (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
%Y A215077 Row sums of A215078, A215079, A215080.
%Y A215077 See also A215083 and A215084.
%K A215077 nonn,nice
%O A215077 0,3
%A A215077 _Olivier Gérard_, Aug 02 2012