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A215078 Triangle of sums of the first k n-th powers multiplied by binomial(n,k), read by rows.

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