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A380747 Array read by ascending antidiagonals: A(n,k) = [x^n] (1 - x)/(1 - k*x)^2.

%I A380747 #8 Feb 03 2025 21:25:07
%S A380747 1,-1,1,0,1,1,0,1,3,1,0,1,8,5,1,0,1,20,21,7,1,0,1,48,81,40,9,1,0,1,
%T A380747 112,297,208,65,11,1,0,1,256,1053,1024,425,96,13,1,0,1,576,3645,4864,
%U A380747 2625,756,133,15,1,0,1,1280,12393,22528,15625,5616,1225,176,17,1
%N A380747 Array read by ascending antidiagonals: A(n,k) = [x^n] (1 - x)/(1 - k*x)^2.
%F A380747 A(n,k) = ((k - 1)*n + k)*k^(n-1) with A(0,0) = 1.
%F A380747 A(n,k) = n! * [x^n] exp(k*x)*(1 + (k - 1)*x).
%F A380747 A(n,0) = A154955(n+1).
%F A380747 A(3,n) = A103532(n-1) for n > 0.
%F A380747 A(n,n) = A007778(n) for n > 0.
%e A380747 The array begins as:
%e A380747    1, 1,   1,    1,     1,     1, ...
%e A380747   -1, 1,   3,    5,     7,     9, ...
%e A380747    0, 1,   8,   21,    40,    65, ...
%e A380747    0, 1,  20,   81,   208,   425, ...
%e A380747    0, 1,  48,  297,  1024,  2625, ...
%e A380747    0, 1, 112, 1053,  4864, 15625, ...
%e A380747    0, 1, 256, 3645, 22528, 90625, ...
%e A380747    ...
%t A380747 A[0,0]:=1; A[1,0]:=-1; A[n_,k_]:=((k-1)*n+k)k^(n-1); Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
%t A380747 A[n_,k_]:=SeriesCoefficient[(1-x)/(1-k*x)^2,{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
%t A380747 A[n_,k_]:=n!SeriesCoefficient[Exp[k*x](1+(k-1)*x),{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten
%Y A380747 Cf. A000012 (k=1 or n=0), A000567 (n=2), A001792 (k=2), A007778, A060747 (n=1), A081038 (k=3), A081039 (k=4), A081040 (k=5), A081041 (k=6), A081042 (k=7), A081043 (k=8), A081044 (k=9), A081045 (k=10), A103532, A154955, A380748 (antidiagonal sums).
%K A380747 sign,easy,tabl
%O A380747 0,9
%A A380747 _Stefano Spezia_, Jan 31 2025