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%I #15 Jan 27 2023 06:42:27
%S 0,0,1,0,2,5,0,3,27,36,0,4,102,392,354,0,5,330,2760,6500,4425,0,6,975,
%T 15880,73350,123090,67171,0,7,2709,81060,654500,2033325,2637327,
%U 1200304,0,8,7196,381808,5064780,25926824,59992660,63259168,24684612,0,9,18468,1696464,35574840,281668590,1034305524,1896003648,1681960464,574304985,0,10,46125,7208880,232816500,2740317300,14981494710,42457884000,64240088580,49143419250,14914341925
%N Triangle of sums of the first k n-th powers multiplied by binomial(n,k), read by rows.
%C If one starts the sum at j=0, the initial term T(0,0) is 1.
%H Vincenzo Librandi, <a href="/A215078/b215078.txt">Table of n, a(n) for n = 0..1000</a>
%F T(n,k) = binomial(n,k)*sum(j^n, j=1..k)
%e 0
%e 0 1
%e 0 2 5
%e 0 3 27 36
%e 0 4 102 392 354
%e 0 5 330 2760 6500 4425
%e 0 6 975 15880 73350 123090 67171
%e 0 7 2709 81060 654500 2033325 2637327 1200304
%p A215078 := proc(n,k)
%p binomial(n,k)*add(j^n,j=1..k) ;
%p end proc:
%p seq(seq(A215078(n,k),k=0..n),n=0..10) ; # _R. J. Mathar_, Jan 27 2023
%t Flatten[Table[Table[Sum[j^n, {j, 1, k}]*Binomial[n, k], {k, 0, n}], {n, 0, 10}], 1]
%Y Binomial convolution of A215083.
%Y Cf. A215077 (row sums), A031971 (diagonal)
%K nonn,tabl
%O 0,5
%A _Olivier Gérard_, Aug 02 2012
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