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A089000 Square table, read by antidiagonals, of coefficients T(k,n) (row k; column n) defined by: T(k,n) = k*T(k,n-1)+ n; T(k,0) = 0.

%I A089000 #13 Apr 19 2025 06:11:22
%S A089000 0,1,0,2,1,0,3,3,1,0,4,6,4,1,0,5,10,11,5,1,0,6,15,26,18,6,1,0,7,21,57,
%T A089000 58,27,7,1,0,8,28,120,179,112,38,8,1,0,9,36,247,543,453,194,51,9,1,0,
%U A089000 10,45,502,1636,1818,975,310,66,10,1,0
%N A089000 Square table, read by antidiagonals, of coefficients T(k,n) (row k; column n) defined by: T(k,n) = k*T(k,n-1)+ n; T(k,0) = 0.
%F A089000 T(k, n)= (k^(n+1)- (k-1)*n - k)/(k-1)^2. T(k, n) = Sum(j, 0<=j<=n; j*k^(n-j)).
%t A089000 Unprotect[Power]; 0^0=1; T[n_,k_]:=Sum[j*k^(n-j),{j,0,n}]; Table[T[n-k,k],{n,0,10},{k,0,n}]//Flatten (* _Stefano Spezia_, Apr 19 2025 *)
%Y A089000 Rows begin:
%Y A089000 {0, 1, 2, 3, 4, 5, 6, 7, 8, ...}:see A001477
%Y A089000 {0, 1, 3, 6, 10, 15, 21, 28, ...} : see A000217
%Y A089000 {0, 1, 4, 11, 26, 57, 120, 247, 502, ...} : see A000295
%Y A089000 {0, 1, 5, 18, 58, 179, 543, 1636, ...} : see A000340
%Y A089000 {0, 1, 6, 27, 112, 453, 1818, 7279, ...} : see A014825
%Y A089000 {0, 1, 7, 38, 194, 975, 4881, 24412, ...} : see A014827
%Y A089000 {0, 1, 8, 51, 310, 1865, 11196, 67183, ...}: see diagonals of triangle A088990
%Y A089000 Diagonal begin:
%Y A089000 {0, 1, 4, 18, 112, 975, 11196, ... } :see A062805
%Y A089000 {0, 1, 5, 27, 194, 1865, ...} : see A023811
%Y A089000 Column {3, 6, 11, 18, 27, 38, 51, ...} : see A010000
%K A089000 easy,nonn,tabl
%O A089000 0,4
%A A089000 _Philippe Deléham_, Nov 02 2003